Clear[H]

H[s_][x_, y_] :=
 f[x] + Piecewise[{{-1, s === "-"}}, 1]*g[y] /; s === "+" || s === "-"

{H["+"][x, y], H["-"][x, y]}

(*  {f[x] + g[y], f[x] - g[y]}  *)

Clear[H]

H[s_][x_, y_] := f[x] + Sign[s]*g[y] /; Abs[s] === 1

{H[1][x, y], H[-1][x, y]}

(*  {f[x] + g[y], f[x] - g[y]}  *)

Needs["Notation´"]

Symbolize[H±]  (* Be careful to use the Writing Assistant palette or other formatting guides s.t. the definition actually displays like you want it to and not Subscript or similar means *)

H±[x_, y_] := f[x] ± g[y] (* Same caveat about typesetting applies here as well *)

In[1] := H±[1,2]
Out[1]:= f[1]±g[2]

h[op_Symbol][x_, y_] := op[f[x], g[y]]

{h[Plus][m, n], h[Subtract][m, n], h[Times][m, n]}
(* {f[m] + g[n], f[m] - g[n], f[m] g[n]} *)

k[op_Symbol /; MemberQ[{Plus, Subtract}, op]][x_, y_] := op[f[x], g[y]]

{k[Plus][m, n], k[Subtract][m, n], k[Times][m, n]}
(* {f[m] + g[n], f[m] - g[n], k[Times][m, n]} *)

(h:SubPlus|SubMinus)[H] ^:= With[
    {hh=Replace[h, {SubPlus->Plus, SubMinus->Subtract}]},
    hh[f[#1], g[#2]]&
]

SubPlus[H][x, y]
SubMinus[H][2, 3]