I asked similar question before, after some modification, I have a new question. Suppose $M^{n\geq 4}$ is a connected compact smooth manifold with connected nonempty boundary. Suppose $i_*: H_1(\partial M)\rightarrow H_1(M)$ is not injective, for simplicity, assume that $\dim (\ker(i_*))=1$, that means, there is 1-cycle (choose a loop $C$) in $\partial M$ which is nontrivial in $\partial M$ but trivial in $M$, assume it is boundary of 2-dimensional embedded submanifold $D$ (let's say $D$ is a 2-disc) in $M$, now denote the small neighbourhood of $D$ in $M$ by $U_\epsilon(D)$ ($\epsilon$ can be arbitrarily small), now I am wondering if new homomorphism $$j_*: H_1(\partial(M\setminus U_\epsilon(D)))\rightarrow H_1(M\setminus U_\epsilon(D))$$ is injective now?

What I have tried: suppose $n=4$, the neighbourhood of $C$ in $\partial M$ is a solid torus, and any circle in the kernel of $i_*$ is homologous to $C$, thus homologous to a circle on the boundary of the solid torus, which retracts along the new boundary $\partial (M\setminus U_\epsilon(D))$.