We introduce a new verification protocol for measurement-only blind quantum computation where the client can only perform single-qubit measurements and the server has sufficient ability to prepare a multi-qubit entangled states. Previous such protocols were limited by strong assumptions about the client's quantum devices. We remove these assumptions by performing self-testing procedure to certify the initial entangled state prepared by the server as well as the operation of the client's quantum devices. In the case of an honest server and client's devices, the protocol produces the correct outcome of the quantum computation. Given a cheating server or malicious quantum devices, our protocol bounds the probability of the client accepting an incorrect outcome while introducing only modest overhead in terms of the number of copies of the initial state needed that scales as $O(n^4\log n)$, where $n$ is the size of the initial universal resource.