Binary and ternary congruences involving intervals and sets modulo a prime

Let $s$ be a fixed positive integer constant, $\epsilon$ be a fixed small positive number. Then, provided that a prime $p$ is large enough, we prove that for any set $\{\mathcal{M}\subseteq {\mathbb{F}}_p^{\ast }$ of size $\mathrm{\mid }\mathcal{M}\mathrm{\mid }=\lfloor p^{14\mathrm{/}29}\rfloor$ and integer $H=\lfloor p^{14\mathrm{/}29+\epsilon }\rfloor$, any integer $\lambda$ can be represented in the form $$\frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}\equiv \lambda p,$$ with $$m_i\in \mathcal{M},1\le x_i\le H,i=1,2,3.$$ When $s=1$ we show that for almost all primes $p$ the following holds: if $\mathrm{\mid }\mathcal{M}\mathrm{\mid }=\lfloor p^{1\mathrm{/}2}\rfloor$ and $H=\lfloor p^{1\mathrm{/}2}(\log p{)}^{6+\epsilon }\rfloor$, then any integer $\lambda$ can be represented in the form $$\frac{m_1}{x_1}+\frac{m_2}{x_2}\equiv \lambda p,$$ with $$m_i\in \mathcal{M},1\le x_i\le H,i=1,2.$$