Is the following extension of finite state automata studied?
Assuming $k$ can be any integer, then this can be formalized as a blind one-counter automaton. Usually these automata accept on final state when its counter is zero, but we can easily model your acceptance type if you allow $\epsilon$ transitions (that do not consume input). If I am not mistaken, like with finite state automata, one can get rid of the $\epsilon$, but that is a non-trivial result.

There are several types of one-counter automata. In the most general form they are allowed to test whether the value of the counter equals zero. 
The languages they accept are a strict subset of the context-free languages. 

The model you are probably looking for is called blind, it cannot test for zero, except as the final test for acceptance at the end of the computation.