How to distributively compute a secret of that form $\mathsf{s=sk\cdot g^a}$? Is there a standard way to distributively compute secrets shares such that any $t+1$ combination out of $n$ of them constructs the secret $\mathsf{s=sk\cdot g^a}$, for a generator $\mathsf{g} \in \mathbb{Z}_p$ and $\mathsf{sk,a}$ chosen uniformly at random from $\mathbb{Z}_p$
I assume that your question relates to sharing a secret specifically of this form. Furthermore, you want to guarantee this even if the dealer is not trusted and may try to share a secret of a different form. Otherwise, you could just use any secret sharing (or verifiable secret sharing) scheme.

However, note that $s$ as you wrote actually does not have any special form. Specifically, since $a$ is uniformly random, $sk\cdot g^a$ is just a uniformly random element in $\mathbb{Z}_p$. Thus, as you have worded your question, you can use any secret sharing scheme for which the domain of the secret is $\mathbb{Z}_p$.