Through careful language any religious leader can serve like advertising in promoting public support for a policy. I personally disapprove of this, since religion is a practice of personal Faith, and should not be 'political', though beginning with England's separation from the domain of the Vatican under the Catholic Church Queen Elizabeth I, initiating the Book of Common Prayer, managed to craftily utilize Church of England leaders to support policies favourable to Her Reign. Thus I am not in favour of this practice, though if persons are not literate nor politically informed, religious outlets are often the source of public policy information cultivating the electorate.
A sociologist in Kiev, Ukraine, couselling adults and youths out of prostitution, often which they were abducted/deceived into, explained the religious and political cult adoration of western societies for me. She had been a doctoral student under the Soviet system and then emerged as an NGO director after the fall of communism working with the U.N. and E.U. in Ukraine. She explained that the eastern and western cognitive styles have generationally evolved forced to 'adore'/worship a 'Jesus' image. In communism 24 hours a day people saw 'Stalin' every where, like Jesus on the walls of churches in the west. When communism came down so did the picture of Stalin on the walls, but the cognitive style remained, as in the west when people stop worshiping the Jesus on the wall - this genetic predisposition to iconocize continues, needing to fulfill a mental silhouette, a mental void, which is easily satisfied by any pimp or politician posing as a 'Jesus/Stalin' figure for both the east and west. In bringing youth out of prostitution pimp-cults she knew she had to find a mental image to replace the pimp for them. They are too afraid to 'loose their religion' so they keep open their mental void filling, and visually following, another image - in Soviet studies it's called a 'cult of personality, though the Soviets failed to erase the DNA for religious/cult worship. In reality, Christianity is based on image worship, whereas Judaism and some forms of Islam basically prohibit or censor image worship, and in an age of visuals in media, I suspect the best solution is to not watch too much media and instead listen, while ensuring that those who get visual time in media are worthy of a cult worship following - images imprinted on the mind are forever https://www.youtube.com/watch?v=xwtdhWltSIg
In Crimea Ukraine/Russia and in Kiev, Ukraine there is a sizeable Muslim population among the Turkish Tatar of Crimea who under Stalin were deported to Uzbekistan, then under President Bush the first (early 1990's) were returned to their 'Homeland' of Crimea. Thus in understanding how to deprogram propaganda, prostitution and illegal human trafficking victims I also had to understand that they were victims of politically motivated mostly Clinton era (post-Chernobyl) western dehumanization policies and understand their inherited biological and religious cognitive structures and cognitive styles. Thus I really studied the Koran, as well as the Torah and the writings of Lenin and Engels. The greatest threat, I found through Ukrainian and Russian cognitive scientists, is that American English, or slang, is used by the mafia to lure eastern European students (Muslim or not), into sex slavery. For instance they are promised work in "writing" or "nursing", then the mafia of western nations tell them they are accommodating a 'pen' (a 'John' in prostitution) or a 'patient' (sex client), thus they 'work' 'writing' or 'nursing'. Not just the Al-Kaeda celebrated 9-11 in Ukraine - hachoo! some 'dust' in nyc https://www.youtube.com/watch?v=28qU_DcoqBs
In the context of the Philippines, which is a massively Christian country, the incorporation of religion and/or spirituality into the restorative justice policies of the state have played a vital role in the rehabilitation of former drug addicts, juvenile delinquents and other law offenders. Pastors and priests work hand-in-hand with social workers in implementing restorative policies (e.g. Juvenile Justice Act which puts emphasis on social welfare interventions) in order to put the subjects into their former state (or at least close to their former state) and to help them avoid recidivism. And it has apparently yielded positive results. So though the separation of the church and the state is a thoroughly logical solution to conflicts, there are still ways to reconcile the roles of both when it comes to certain human issues.
Religion can play a positive role in the formulation and implementation of good public policy if it has or support a social ethics that promotes the dignity of each human individual, social justice, human rights and democracy.
Lately, religion is understood differently. Nevertheless, if taken from basic understanding of what religion (relegare - to bind or bridge the gap) is: bridging the gap of relationship with the source/divine/God, etc. which should consequently results in the common good of the society. In addition, the fundamental of civil society is the well-being of all. Human dignity as well as human rights constitute what should be the aim of good policy. Hence, separating religion from society is like repressing sexual urge. John Gray puts it aptly, “those who demand that religion be exorcised from politics think this can be achieved by excluding traditional faiths from public institutions; but secular creeds are formed from religious concepts, and suppressing religion does not mean it ceases to control thinking and behavior. Like repressed sexual desire, faith returns, often in grotesque forms, to govern the lives of those who deny it.”
The role of Religion was important in Greece, Egipt, the Roman Empire, the Byzantine Empire, Medieval Christian kingdoms, the United Kingdom (after XVI Century), etc. However, there are some features in Christian Tradition, not allways understood/followed by christian rulers/actors in past, that make things more complex: a) Christian tradition thinks this world is incomplete (and policy, economy, and all human activites and edeavours are seen as a way of solving problems, overcome difficulties and completing what is imperfect, that is, the cosmical vocation of human beings); b) Christian tradition has some very general principles for the world, because Christ is the Key of creation, the Word of God became flesh and lived with men and women, but these principles can be followed in many different ways (one of them is the importance of each human person, even if having no credit card); c) some religions were a product of society and culture, a way to address the spiritual needs of men and women, but christians believe that Christian religion came from God's initiative, gathering human persons in one people (it's social dimension came from above) and transcends the society where it lives (there are christian communities in very different societies, and none of them can be situated in the same level with social and political organizations). This means that: 1. Christian Tradition sees policy and all human activities like a mission in the world; 2. Christian solutions for society are seen in a non absolute way (normally, there is room for different christian responses to the same problem, and there is room for better/worse solutions as time goes by); 3. There is a healthy separation between Church and human organizations of any kind (political, economical, social, etc); in Catholic Tradition is called "autonomy of creation" (see Pastoral Constitution "Gaudium et spes" of 2nd Vatican Council). Politicians can use religion for worldly purposes and religious ministers/actors can use policy for religious purposes, but, if you believe religion is not a (by-)product of society and there is a healthy separation between "organized" religion and social organizations, you may conclude that both are an abuse. If you want to overcome this kind of abuses, you may follow Hobbes (or Karl Marx) in supporting a very limited influence of religion in the public sphere of society; or you may follow amish way of life, setling aside from the common world, and preventing yourself of being influenced from other than your amish faith. I'm afraid both are trying to absolutize what can not be absolutized: this economic (Marx), social and political world (Hobbes) and an unique religion based social solution (amish). They cannot be presented as christian solutions. Answering the questions above: religion has some influence in public policy, but it's difficult to caracterize this influence for two main reasons. First, religion is not "designed directly for" public policy as such; second, public policy is influenced by many actors and elements, and religion is not the main/only one. However, history tells us that Christian Tradition had strong influence in, for instance, considering the value of human freedom and human person for society. Some useful reading: Joseph Ratzinger's book, "Truth and Tolerance: Christian Belief and World Religions" (https://www.amazon.com/Truth-Tolerance-Christian-Belief-Religions/dp/158617035X), and Joseph Ratzinger-Jurgen Habermas, "The Dialectics of secularization: On Reason and Religion" (https://www.amazon.com/Dialectics-Secularization-Reason-Religion/dp/1586171666/ref=pd_lpo_sbs_14_t_0?_encoding=UTF8&psc=1&refRID=D9C3ESTT3D3WGC1KX1P0). You will find an outstanding study by Martin Ronheimer on the relationship between Christian Religion and politics, in chapter XII (Chriatianity and secularity: Past and Present of a Complex Relationship) of his book "The Common Good of Constitutional Democracy" available on-line (https://books.google.it/books/about/The_Common_Good_of_Constitutional_Democr.html?id=O1LrIBBMRJIC&redir_esc=y). In spanish you may find A. Bellocq's study: "El papel de la Iglesia en la política a la luz de las enseñanzas de Benedicto XVI" (http://revistas.um.edu.uy/index.php/revistahumanidades/article/view/61/53), and in italian (http://vanthuanobservatory.org/_files/bollettini/allegati/bellocq.pdf).