Consider the long exact sequence on homology (coefficients in $\mathbb{Z}$) $$\to H_1(\partial M)\overset{i}{\to} H_1(M)\to H_1(M,\partial M) \to H_0(M) \to $$$$\to H_1(\partial M)\overset{i}{\to} H_1(M)\to H_1(M,\partial M) \to H_0(\partial M) \to $$ You are looking for $H_1(M)/i_\ast(H_1(\partial M)) \cong im\{ H_1(M)\to H_1(M,\partial M)\} \cong ker \{ H_1(M,\partial M)\to H_0(M)\}$$H_1(M)/i_\ast(H_1(\partial M)) \cong im\{ H_1(M)\to H_1(M,\partial M)\} \cong ker \{ H_1(M,\partial M)\to H_0(\partial M)\}$ by the exactness of the sequence. If $H_1(M)\cong \mathbb{Z}^n\oplus T$, $T$ torsion, then $H_2(M)\cong \mathbb{Z}^{n-1}$ (the rank $n-1$ follows from Euler characteristic). By universal coefficients, we have a short exact sequence $$0\to Ext(H_1(M),\mathbb{Z})\to H^2(M) \to Hom(H_2(M),\mathbb{Z})\to 0.$$ One computes $Ext(H_1(M),\mathbb{Z})\cong T$, $Hom(H_2(M),\mathbb{Z})\cong \mathbb{Z}^{n-1}$ (see p. 195 of Hatcher), so that $H^2(M)\cong \mathbb{Z}^{n-1} \oplus T$. By Lefschetz duality, $H^2(M)\cong H_1(M,\partial M) \cong \mathbb{Z}^{n-1}\oplus T$. So you are looking for the torsion in $$ker\{ H_1(M,\partial M)\to H_0(M) \} \cong ker\{ \mathbb{Z}^{n-1}\oplus T \to \mathbb{Z}\},$$$$ker\{ H_1(M,\partial M)\to H_0(\partial M) \} \cong ker\{ \mathbb{Z}^{n-1}\oplus T \to \mathbb{Z}^c\},$$ which is clearly isomorphic to $T$.
