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What you can do is choose a monomial ordering on the polynomial ring such that $y$ is the largest variable, so that in the multivariate division algorithm computing the remainder after division by $y^2 - (x^3 + x)$ will amount to substituting $y^2 = x^3 + x$:

sage: R.<x,y> = PolynomialRing(GF(43), order='invlex')
sage: (y^2 + x*y + 1).reduce([y^2 - (x^3 + x)])
x*y + x^3 + x + 1

Here you could also choose e.g. R.<y,x> = PolynomialRing(GF(43), order='lex').