p[y_, t_, k_] := ((1 - E^(-((-1 + y)^2/t))) t)/(-1 + y)^2 -
                ((1 - E^(-((1 + y)^2/t))) t)/(1 + y)^2 - k

FindRoot[p[y, t, 0.5], {y, 1}]

During evaluation of In[3]:= Power::infy: Infinite expression 1/0.^2 encountered. >>

During evaluation of In[3]:= Infinity::indet: Indeterminate expression 0. t ComplexInfinity encountered. >>

During evaluation of In[3]:= FindRoot::nlnum: The function value {Indeterminate} is not a list of numbers with dimensions {1} at {y} = {1.}. >>

(* FindRoot[p[y, t, 0.5], {y, 1}] *)

p[y_, t_, k_]:= ((1 - E^(-((-1 + y)^2/t))) t)/(-1 + y)^2 -
                ((1 - E^(-((1 + y)^2/t))) t)/(1 + y)^2 - k

t = Table[i, {i, 0, 5}]
NSolve[p[y, t, 0.5], y]

ContourPlot[p[y, t, 0.5], {y, 0, 1}, {t, 0, 1}, PlotLegends -> Automatic]

Table[y /. FindRoot[p[y, i, 0.5], {y, 0.5}], {i, 0.1, 1, 0.1}]

Table[y /. FindRoot[p[y, i, 0.5], {y, 0.5 i}], {i, {1, 2.1, 3}}]