That looks like a Holset HX 400 variable geometry turbo. (VGT) see pic.
If this is correct, that shaft is what moves the turbine blades to achieve the variable geometry.
So with the shaft broken, the turbo is now a fixed geometry turbo and apparently it is in a position mostly open which would develop full power but should be somewhat laggy on spoolup.
Im not familiar enough to know if running it this way will cause any damage or not. If the blades are fixed and not moving it would be the same as a fixed geometry turbo but if somehow they are continually moving or flapping that may cause catastrophic failure of the turbine
Best thing to do is repair or replace it for sure.