yes, just include them in the model, making sure the package recognizing them as categorical variables (though I don't like calling this "controlling" since it is not doing that). A useful reference is: https://meehl.umn.edu/sites/meehl.umn.edu/files/files/084nuisancevariables.pdf

It is not clear how your design is. I guess neither of your categorical variables is a dependent variables.

I guess you want to run a multiple regression with your categorical variables as predictors. To 'control' for the mutual effects, you could take into account some possible interactions, e.g. between Gender and Income.

@Partha Yes categorical variables can be used as a covariate in ANCOVA.

I would run a stepwise multiple regression, or an 'all possible subsets' regression, in order to find out which categorical variables are most effective/relevant. I would follow up such a regression with a cross-validation. There are numerous books and websites on these subjects. I personally liked Jared Lander's book: "R for Everyone", wich has excellent chapters on these matters.

https://www.researchgate.net/post/Is-it-possible-and-acceptable-to-use-a-categorical-covariate-in-ANCOVA

A similar question is here.

Daniel Wright, Jos Feys and Imran Anwar, I have a question:

If we use a categorical variable as the covariate, does it change the name/title of the test/method?

In my opinion you can not, for example, use Gender as a covariate. This would imply there is something wrong with Gender and you have to correct for it.