Let $2^* = \frac{2n}{n-2}$ where $n$ is the dimension of a compact closed manifold $M$.
We get from the Sobolev/Poincare inequality the identity $$\lVert u \rVert_{2^*} \leq C\lVert \nabla u \rVert_2 + \lVert \overline{u}\rVert_{2^*}$$ where the norms are $L^p$ norms. Now we can estimate $$\lVert \overline{u}\rVert_{2^*} \leq C\lVert u \rVert_1$$ and we find $$\lVert u \rVert_{2^*} \leq C\lVert \nabla u \rVert_2 + \lVert {u}\rVert_{1}.$$ I am wondering is it possible to get something better than this in the sense that instead of the last term we have something like $\lVert u \rVert_{1}^p$ where $p < 1$?
The second term is causing me problems which I would avoid if I worked on a bounded domain with zero boundary conditions since I just use the nice Poincare inequality there.
No, you can't have $p<1$ there. Take the constant function $u\equiv \lambda$. Your inequality becomes $$\|\lambda\|_{2^*}\le \|\lambda\|_1^p$$ which (if $p< 1$) fails when $\lambda$ is large enough.
When you imagine an inequality you'd like to be valid, consider how it scales when $u$ is multiply by a positive number, or (when working on vector spaces) when the argument of $u$ is rescaled.