Here is something that's valid in the stable range.
If $M$ and $N$ are closed $n$-manifolds, there is a cofibration sequence $$ S^{n-1} \to M_0 \vee N_0 \to M\sharp N $$ where $M_0$ denotes the effect of deleting a point from $M$.
If $M$ and $N$ are $r$-connected, then so is the connected sum. The Blakers-Massey excision theorem then implies an exact sequence $$ \pi_k(S^{n-1}) \to \pi_k(M_0 \vee N_0) \to \pi_k(M\sharp N) \to \pi_{k-1}(S^{n-1}) \to \cdots $$ as long as $k \le n-2+r$.
Furthermore the map $M_0 \vee N_0 \to M_0 \times N_0$ is $(2r+1)$-connected, so if $k \le 2r$ we get $\pi_k(M_0 \vee N_0) = \pi_k(M) \oplus \pi_k(N)$.
Assembling this, we have an exact sequence $$ \pi_k(S^{n-1}) \to \pi_k(M) \oplus \pi_k(N) \to \pi_k(M\sharp N) \to \pi_{k-1}(S^{n-1}) \to \cdots $$ which is valid for $k \le 2r$, $r \le n-2$.
Added Later
I just realized one could simply note that the cofiber sequence gives a long exact sequence on stable homotopy $$ \pi_k^{st}(S^{n-1}) \to \pi_k^{st}(M_0) \oplus \pi_k^{st}(N_0) \to \pi_k^{st}(M\sharp N) \to \pi_{k-1}^{st}(S^{n-1}) \to \cdots $$ and then if $M$ and $N$ are $r$-connected with $k \le 2r$ and $r\le n-2$ we can use the Freudenthal suspension theorem to identify the stable groups with the corresponding unstable ones. This gives a more elementary argument.
Here's a special case: when $M$ and $N$ are framed, so is $M\sharp N$ and the connecting map in the exact sequence splits to give a splitting $$ \pi_k(M\sharp N) = \pi_k(M) \oplus \pi_k(N) \oplus \pi_{k-1}(S^{n-1}) $$ (assuming the constraints on $k,r$ and $n$).
