Assessing and acknowledging publication bias is a good starting point; there are also some methods of assessing the impact of publication bias on the overall results, see this paper for guidance Article Publication Bias: The Elephant in the Review
I recommend applying a Duval and Tweedie's Trim & Fill correction to the overall effect size. It is the only procedure that attempts to make a correction for the presence of publication bias.
Apart from 'Trim and Fill' - which is the easiest approach - there are more complex statistical approaches to specifically adjust for the probability that a finding is published, by using weights in the main model (i.e. selection method approaches).
The book "Publication Bias in Meta-Analysis: Prevention, Assessment and Adjustments" by Hannah R. Rothstein, Alexander J. Sutton, Michael Borenstein, Wiley: 2005 contains additional details on this subject.