For my current project I need to compare means of four groups by one-way ANOVA. In order to test whether my data come from normal distribution or not, I have checked each group for normality by Shapiro-Wilk test and now I have four p-values, i.e. one p-value per group. Should I apply Bonferroni correction to these p-values?
Bonferroni is used to control false discoveries (Type I errors). Your 4 p-values, if I'm interpreting your question correctly, are from assumption tests, not from tests to demonstrate the significance of your discoveries, and therefore don't call for Bonferroni correction. I doubt that you are trying to demonstrate non-normality, or that you would claim a "discovery" for detecting non-normality. In fact, the goal of assumption tests is typically NONSIGNIFICANCE, not significance. Therefore, there is no reason to apply Bonferroni correction to assumption tests under typical circumstances.
In Shapiro-Wilk test, the null hypothesis is that the sample come from a normal distribution. You can use your p-value to reject the null hypothesis. If your tests show large departure from normality assumptions, ANOVA might not be an appropriate model.
There's no reason to apply Bonferroni correction here. You either go ahead with ANOVA if you believe the assumptions are met or choose another model if the assumption are not met. The normality test only guide you if you want to perform the the model.
You should check your QQ plot.