Suppose I have a study with a response variable $y$ and two explanatory variables $x_1, x_2$. I do a regression such as
lm(y~x1) and get a p-value of $p_1'$ for $x_1$. Then I control for $x_2$ and run
lm(y~x1 + x2) and get $p_1''$ for $x_1$ and $p_2''$.
Question 1 How would I interpret the following cases in terms of if my model benefits from adding the covariate?
Question 2 Will the answer extend to more explanatory covariates?
Question 3 What role does $p_2''$ play in my analysis/interpretation?