Homomorphic Evaluation of Pseudo Random Function
There is an Eurocrypt'15 paper published by Albrecht et al. which creates a cipher minimizing the number of multiplications needed to produce an encryption. This cipher is called LowMC and it can also be used as a PRF with almost same parameters as in the block-cipher - check out LowMC section in this paper.

Fortunately, there is open-source code which evaluates LowMC homomorphically with HElib. To evaluate the cipher homomorphically is to compute $\mathsf{Enc}_{\mathsf{k}}(\mathsf{LowMC}(x))$ from $\mathsf{Enc}_\mathsf{k}(x)$.

So you can plug in the PRF parameters given by Grassi et al. (for 128 bit PRF security) in the script provided by Albrecht and there you have it: LowMC becomes a PRF and your question is solved - compute $\mathsf{Enc}_\mathsf{k}(\mathsf{PRF}(x))$ from $\mathsf{Enc}_\mathsf{k}(x)$. And I think this is the state of the art for computing PRFs in HE land.