Let $M^{n\geq 4}$ be a compact connected manifold with connected boundary $\partial M$. Suppose $$i_*: \pi_1(\partial M)\rightarrow \pi_1(M)$$is not injective, it means that there is non-contractible loop in $\partial M$ which is contractible in $M$. And also we can assume this loop spans a disc $D$ in $M$, denote the small neighbourhood of $D$ in $M$ by $U(D)$. I am wondering if $$i_*:\pi_1(\partial (M\setminus U(D)))\rightarrow \pi_1(M\setminus U(D))$$ is injective now. I am just trying to make the non-jectivity disappear by deleting it. It looks right to me but I don't know how to rigorously explain it. Or maybe I am wrong and there is counterexample. Thanks for your help.
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The comments have shown that you approach does not work.
If $i_* : H_1(\partial M) \to H_1(M)$ is not injective, you cannot deduce that $i_* : \pi_1(\partial M) \to \pi_1(M)$ is not injective.
A loop in $\partial M$ which is contractible in $M$ need not span a disk in $M$. Take for example $M = D^2$. Then $\partial M = S^1$. Any loop in $S^1$ is contractible in $D^2$, but in general it does not span a disk $D$ in $D^2$. You cannot even find a homotopic loop for which it is true unless the loop has degree $\pm1$.
In the above eaxmple, you have $D = D^2$ and thus $U(D) = D^2$.