I have a bit of time, so I'll be explicit. This construction assumes $\pi: G\to G/H$ is a fiber bundle, and that $G/H$ is paracompact Hausdorff (and so admits partitions of unity). This should cover many examples that occur "in nature." My comments below the original question also do the case where $H$ is compact, which I think need not be contained in this case.

I'll need the following lemma.

**Lemma.** Under the given assumptions, there is a continuous function $s: G\to \mathbb{R}$ such that $s$ has compact support when restricted to each fiber of $\pi$, and such that for each $g\in G$, $\int_H s(gh) d_Hh=1$.
**Proof.** If $\pi: G\to G/H$ is trivial as a principle $H$-bundle (that is, it admits a section $t: G/H\to G$), this is easy; namely, pick any continuous function $s': H\to \mathbb{R}$ with compact support, satisfying $\int_H s'(h) d_Hh=1$. Let $s(x)=s'(t(\pi(x))^{-1}x)$.

Now if $\pi: G\to G/H$ is a fiber bundle, we may cover $G/H$ by open $U_i$ such that the bundle is trivial over each $U_i$. By the previous paragraph, we may choose $s_i: \pi^{-1}(U_i)\to \mathbb{R}$ with compact support on each fiber of $\pi$, and whose integral over each fiber equals $1$. Now by assumption we may choose a partition of unity $\{\phi_j\}$ subordinate to the cover $\{U_i\}$. Let $s=\sum_{i,j} s_i\cdot (\phi_j\circ \pi)$. $\Box$

We now construct a right inverse $D$ to $P$. Namely, for $f$ a compactly supported continuous function on $G/H$, let $D(f)=s\cdot (f\circ \pi)$. It is clear that $P(D(f))=f$; one need only check that $s\cdot (f\circ \pi)$ has compact support, which I leave as an easy exercise (again using that $\pi$ is a fiber bundle).