Implementing addition for a binary counter
Well, surely there is.

As a first thing I would convert $k$ to its binary representation $\text{bin}(k)$. If $c$ is the current value of your counter then in order to perform $\text{add}(k)$ just update the counter by setting it to $\text{bin}(c)+\text{bin}(k)$.

If you don't know how to add binary numbers efficiently, think how you would have done it using pen and paper.

For small $k\in O(\log c)$ the repeated increment might be efficient though.