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The sphere minus 4 points is doubly covered by the torus minus 4 points. This double cover gives a representation of $\pi_1(S^2\setminus\mbox{4 points})$ in the symmetric group on two letters. Namely, each of your generators $g,h,x,w$ gives a transposition. The kernel of this representation is the image of the fundamental group pf the torus minus 4 points. (Usually there is an ambiguity coming from the choice of base point in the covering space, but not in this example, since the covering is regular.) The kernel is formed by all words containing an even number of letters.