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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 $k \le 2r$, $r \le n-2$.

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 $n$).

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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 $r k \le n-4$ 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+1$2r$, $r \le n-4$n-2$. 3 added 1 characters in body; added 13 characters in body Here is something that's valid in the stable range. If$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$r \le n-4$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 vaid valid for$k \le 2r+1$,$r \le n-4\$.

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