Thanks for adding the additional example above! I just realized that it was only a rounding issue which led to me seeing 2 different p-values. Sorry for my confusion :)

Yes, I am using wald.test from aod. I specified term as the location of the covariate in the model (e.g. Terms=2 since the covariate gender is the first covariate after intercept in my model).

Thanks for the explanation! I was wondering why the wald p-value is not the same as the p-value from the wald test. I saw there is the possibility to use wald.test(b=coef(model), Sigma=vcov(model), Terms=xx) in R. I used it to check about the p-value from the wald test (with the null hypothesis being that the coefficient of gender is zero), and it was different from the p-value for the covariate gender as shown in the GLM summary.

Thanks for the input, I just updated the question to include that

Thanks for the comment! I am using the R, no, the p-value is different in two cases