Question: How to evaluate the double integral?

I would like to plot the following singular double integral, but I cannot due to singularities...

 

where x>0, t=0.2 and m=0.2.

I defined f(y) function as f:=y->exp(-(y-4.68)^2/0.4):

I attached my file:
1st_try.mw

Thank you !

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