In most New Keynesian models, can monetary shock affect technology shock also?
No it usually does not.

There is an intuitive reason why: We interpret the technology shock as a residual, the unexplainable part in TFP changes. If we knew that there was a specific causal relationship between monetary policy and future TFP, that part of future TFP would now was explained. 

It's true that the MP shock is excluded in standard Solow Regressions, but if you believe there is a causal effect, you need to regress it out of it. Since after all, we understand $A_t$ to be independent.

Note that still, implicitly, through its effects on capital accumulation (which does not exist in standard NK models), monetary policy should impact future TFP. Then, you could make $A_t$ dependent and argue it being a shortcut in the absence of capital.