For fixed $r$ you may expand ${2k-n+a\choose r}$ as a polynomial in $k$ (whose coefficients are polynomials in $n-a$) as ${2k-n+a\choose r}=\sum_{i=0}^r c_i k(k-1)\ldots (k-i+1)$, use $${n\choose k}k(k-1)\ldots (k-i+1)=n(n-1)\ldots(n-i+1){n-i\choose k-i},$$
and apply Vandermonde convolution $$\sum_{k=i}^n {n-i\choose k-i}{n\choose k+a}={2n-i\choose n+a}.$$