I think it would work. PoE 802.3at requires only two pairs to deliver power, with regards to ethernet connection you will only be able to get 100Mpbs as for 1Gbps you need all four pairs. You might need to force the switch port to 100Mbps speed.
Just to be clear. The fact that it would work does not mean that you should do it. Probably replacing the broken cable is the way to go. That way you will be able to have a good performance with the MR56 and avoid other issues.