Supercongruences concerning lacunary sums of Catalan numbers and binomial coefficients

We consider two conjectures of Sun in [22] concerning lacunary sums of Catalan numbers and binomial coefficients. As a conclusion, we completely confirm one of Sun's conjecture and partially confirm the other one. For example, suppose that $p\geq 3$ is prime and $0\leq r\leq p-1$. Then, for any $a\geq 0$, we have \begin{align} \notag S_r(p^{a+2})\equiv S_r(p^a)\pmod{p^{1+a}}, \end{align} where $$S_r(p^a)=\sum_{\substack{0＜ k ＜ p^a \\k\equiv r\pmod{p-1} }}C_k,$$ and $C_k=\binom{2k}{k}/(k+1)$ is the $k$-th Catalan number. Furthermore, when $p=2$, we have \begin{align} \notag S_r(2^{a+2})\equiv S_r(2^a)\pmod{2^{2(1+a)}}. \end{align}