This terms are used interchangeably, however, more specific, relationship is used when study some variables that are no known to have relation to each other but association, when this a relation between two variables (- or + or weak, strong, or no). The influence or effect to see is there any change can we have (unknown: positive or negative, more or less). Impact means that has a strong and profound effect.
In academic writing when we can use association, relationship, effect, influence and impact?
Association & relationship might suggest correlationship i.e. bi-directional between 2 variables whereas effect, influence & impact suggest unidirectional from one variable to another. Regardless which term to use, think author should always operationally define the conceptual framework, research model, hypotheses etc. & better still in pictorial form or ask readers refer to the conceptual framework or research model diagram in the study so that readers really understanding whether a relationship is bi-directional or unidirectional.
In academic writing, the terms you are interested in are oral means of describing statistical relationships. One variable is associated or has a relationship with another when the statistical correlation is statistically significant. To use either term in any other context is misleading. As previously mentioned, there is no directionality in the use of association or relationship.
The use of effect is also related technically to the statistical test used to determine effect size.
Semantically, influence may suggest directionality and should be rarely used unless the research design at least suggests time sequence between the variables. In my opinion, it is less sophisticated to use the term "impact" in academic writing as it suggests a qualitative element that cannot be empirically justified.