For 3x3 and 4x4 matrices, the map A1 x1^k + A2 x2^k is surjective if and only if the nullity of A2 meets conditions depending on n and k, when A1 is invertible.
The images of non-commutative polynomials evaluated on 2×2 matrices over an arbitrary field
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Polynomial Maps with Constants on Matrix Algebra
For 3x3 and 4x4 matrices, the map A1 x1^k + A2 x2^k is surjective if and only if the nullity of A2 meets conditions depending on n and k, when A1 is invertible.