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Let $f\left(𝐩\right)$ be an additive form of degree $k$ with $s$ prime variables $p_1,p_2,\cdots ,p_s$. Suppose that $f$ has real coefficients ${\lambda }_i$ with at least one ratio ${\lambda }_i/{\lambda }_j$ algebraic and irrational. If s is large enough then $f$ takes values close to almost all members of any well-spaced sequence. This complements earlier work of Brüdern, Cook and Perelli (linear forms) and Cook and Fox (quadratic forms). The result is based on Hua’s Lemma and, for $k\ge 6$, Heath-Brown’s improvement on Hua’s Lemma.