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.