I am redoing this question as the first rendition has been a bit of a mess.
Throughout this question, we want to only consider homology with coefficients in $\mathbb{Q}$, so we can ignore issues that arise with torsion.
Let $M$ be a compact orientable $n$-dimensional real manifold with boundary $\partial M$. Lefschetz duality as it can be found in Hatcher (Theorem 3.4.3) states that there are isomorphisms $H^i(M) \cong H_{n-i}(M, \partial M)$ for all $i$, and using the universal coefficients theorem one obtains perfect pairings $H_i(M) \otimes H_{n-i}(M, \partial M) \to \mathbb{Q}$.
What is not covered in any of the sources I've consulted, however, is that these pairings induce a duality (and hence more perfect pairings) between $\operatorname{im} (H_i(M) \to H_i(M, \partial M))$ and $\operatorname{im} (H_{n-i}(M) \to H_{n-i}(M, \partial M))$, which is mentioned in this MO question as well as in Example 3.2.12 of Greg Friedman's Singular Intersection Homology (p. 97). These maps are those that appear in the long exact sequence of the pair, see below.
I have noticed the sketch of a proof in a comment Tom Goodwillie made on the mathoverflow post linked above, which I tried to follow. Let's suppose that $R = \mathbb{Q}$, so we can identify $H^i$ with the dual of $H_i$. Consider the long exact sequence associated to the inclusion $\partial M \hookrightarrow M$: $$ \cdots \longrightarrow H_i(\partial M) \overset{q_i}{\longrightarrow} H_i(M) \overset{p_i}{\longrightarrow} H_i(M, \partial M) \longrightarrow \cdots $$
Then, Lefschetz duality gives an isomorphism $H^i(M) \to H_{n-i}(M, \partial M)$, hence a perfect pairing $H_i(M) \otimes H_{n-i}(M, \partial M) \to \mathbb{Q}$. This obviously restricts to a pairing $b$ between $H_i(M)$ and the image of $p_{n-i}$. It is then claimed that the pairing factors over $(H_i(M) / \operatorname{ker} p_i) \otimes \operatorname{im} p_{n-i}$ and that moreover this is a perfect pairing. I am unable to prove the factorization property.
I have tried showing that $b$ vanishes on $\operatorname{ker} p_i \otimes \operatorname{im} p_{n-i}$, where the first factor of this tensor product can be rewritten as $\operatorname{im}q_i$ by exactness of the long exact sequence. This was where I got stuck. Any help would be appreciated.