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The covTest package is an experimental R package for significance tests with regularized regression (the paper introducing it is here). I haven't used it, but it's by the folks who invented the lasso and who authored the glmnet package for regularized regression in R (glmnet vignette here).
When you have a variable that's a linear combination of other variables (for example, if you have independent variables x, w, and z, but x = aw + bz (where a and b are any real constants other than zero), then x is a linear combination of w and z) you're essentially asking glm to estimate n + 1 regression coefficients with only n equations. But there's no unique solution in that case.
glm drops one of the linearly dependent variables from the regression and returns NA for that coefficient. It has nothing to do with whether there are NA values in your data.
