I am curious if somebody can be helpful concerning the following experimental observation:

There exist two rational sequences $\alpha_0,\alpha_1,\dots$ and $\beta_0,\beta_1,\dots$, both with values in $\mathbb Z[1/3]$ such that $$\sum_{k=0}^{p-1}{2k\choose k}\frac{k^j}{k+1}\equiv \alpha_j+p\beta_j\pmod{p^2}$$ for every prime number $p\equiv 1\pmod 6$ and $$\sum_{k=0}^{p-1}{2k\choose k}\frac{k^j}{k+1}\equiv -(-1)^j-\alpha_j+p\beta_j\pmod{p^2}$$ for every prime number $p\equiv 5\pmod 6$.

(More precisely, the sequences $3^n\alpha_n$ and $3^n\beta_n$ are seemingly integral.)

The sequence $\alpha_0,\alpha_1,\dots$ starts as $$1, 0, -2/3, 4/3, -22/9, 140/27, -14, 1316/27, -17078/81, 87860/81, -1562042/243, 31323292/729, \dots$$ and the first terms $\beta_0,\beta_1,\dots$ are $$0,0,2/3,-2,14/3,-34/3,98/3,-350/3,1526/3,-2622,46634/3,-311734/3,2316158/3, -18920018/3,\dots$$

Let me end by remarking that one has as a special case a similar result when replacing Catalan numbers by central binomial coefficients.

Update: The existence of the sequence $\alpha_n$ is explained by the Zhi-Wei Sun paper, see the answer by dke below.

Experimentally, the quotient sequence $\frac{\beta_n}{\alpha_n}$ (defined for $n\geq 2$) seems to converge very quickly towards $-\frac{4\sqrt{3}\pi}{9}=-2.4183991523\dots$ (the error is smaller than $10^{-78}$ for $n=120$).

The sequence $\frac{\alpha_{n+1}}{\alpha_n}-\frac{\alpha_n}{\alpha_{n-1}}$ converges perhaps (fairly slowly) towards something like $-.72\dots$.