Number of integers represented by families of binary forms III: fewnomials

In a series of papers we investigated the following question: given a family $\calF$ of binary forms having nonzero discriminant and integer coefficients, for each $d\geqslant 3$, we estimate the number of integers $m$ with $|m|\leqslant N$ which are represented by an element in $\calF$ of degree $\geqslant d$. Under suitable assumptions, asymptotically as $N\to\infty$, the main term in the estimate is given by the forms in $\calF$ having degree $d$ (if any), while the forms of degree $>d$ contribute only to the error term. The present text is devoted to fewnomials $$a_0X^{kr}+a_1X^{k(r-1)}Y^k+\cdots +a_{r-1}X^kY^{k(r-1)}+a_rY^{kr}$$ with fixed $r\geqslant 1$ and varying $k,a_0,a_1,\dots,a_r$.