ContourPlot[x^3 - y^2 == 2 y, {x, -1, 1}, {y, -1, .1},
RegionFunction -> (#^2 + #2^3 > 2 # &)]
The following procedure contains additional condition which you've provided by range for RegionPlot: {x, -1, 1}, {y, -1, .1}.
Reduce[x^3 - y^2 == 2 y && x^2 + y^3 > 2 x && Abs[x] <= 1 && -1 <= y <= .1,
{x, y}, Reals]
-1. <= x < 0 && y == -1. + Sqrt[1. + x^3]