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

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