I very much doubt that you can find a formula for the solutions with symbolic parameters. You can try numerically for specific numerical values for the parameters. For example, setting all parameters equal to 1 you get two numerical solutions:
expr = Simplify[g + h /. Thread[{a, b, c, d, x, y} -> 1]]
Plot[expr, {r, 0, 2 Pi}]
NSolve[expr == 0 && 0 < r < 2 Pi, r]
