I have seen their applications in quantum state tomography but not in computation as such.
The weak measurement was primarily built as a tool for tomography for obvious reasons, to estimate and approximate the state statistically with the least disturbance. They do not have much use in the conventional Quantum Circuit model and other formalisms because the measurement and readout are performed after all the gate operations are done on specific qubits, and because you require to extract the maximum information out of the measurement, there isn't any intrinsic requirement of a weak measurement. Also, in all conventional circuits, we have the state and register initialization only once, in the start of the protocol. This is the basic layout of most quantum algorithms in use.
However, there are recent few extended protocols where you require to preserve and re-initialize the qubits successively after readout, there comes the use of weak measurement wrt certain trade-off parameters.
One such article which claims to increase efficiency using weak measurements is (without restarting a whole protocol every time): https://iopscience.iop.org/article/10.1088/1367-2630/13/5/053024/meta