The above can be simplified as follows:

PowerExpand[Log[-(1/c)] + Log[c], Assumptions -> Arg[c] > 0]

which gives:

I Pi

As:

w[z] = Exp[-z^2] (1 + I Erfi[z])

the expected result follows straightforwardly for Im[z] > 0.
Edit:
For plotting and computation, I just found a Resource function contributed by Jan Mangaldan here:
https://resources.wolframcloud.com/FunctionRepository/resources/FaddeevaW/
Being a resource function, it will not be used by the core symbolic integration engine.

Yes you are right , as workaround:

 InverseMellinTransform[
  Integrate[
    MellinTransform[Exp[-x^2]/(c - x), c, s] // 
     PowerExpand, {x, -Infinity, Infinity}, PrincipalValue -> True] // 
   Expand, s, c]


(*(-MeijerG[{{0, 1/2}, {}}, {{0, 0, 1/2}, {}}, -c, 1/2] + 
 MeijerG[{{0, 1/2}, {}}, {{0, 0, 1/2}, {}}, c, 1/2])/(2 \[Pi])*)

Only by solution with MeijerG function.