I am trying to prove the following theorem,
Given a smooth manifold $M$ which has only one end and each of whose finite coverings have only one end, the inclusion map $i: \partial M \rightarrow M$ induces a surjection $i_*: \pi_{1} (\partial M) \rightarrow \pi_{1} (M)$.
There is another theorem, can be found in a paper by Gilles Carron and Emmanual Pedon, Proposition 5.2, page 27, which uses almost the same hypothesis to show that $H^1_0(M,\mathbb{Z})=\left\{0\right\}$. I was trying to construct a non-zero element in $H^1_0(M,\mathbb{Z})$ starting from an element in $\pi_1(M)-\pi_1 (\partial M)$ but I am unable to. Could anyone please tell me if this line of thought appropriate or propose another method?
What you are trying to prove is simply false. (I assume that by "end" you mean an end in the sense of Freudenthal as discussed here.) Take a closed 4-manifold $N$ with simple infinite fundamental group (such things exist). Let $M$ be obtained from $N$ by removing one point and then an open ball. The manifold $M$ has $\pi_1(M)\cong \pi_1(N)$, one boundary component which is the 3-sphere and one end. Then $M$ has no nontrivial finite (connected) covers (since $\pi_1(M)$ is simple and infinite). At the same time, $1=\pi_1(\partial M)\to \pi_1(M)$ is definitely not onto.
I strongly suspect that you have in mind more restrictions on the manifold $M$ and maybe by an "end" you mean something very nonstandard. If this is the case, you should spell it out.