The space $P(k) \backslash G(k)$ has a discrete set of representatives in $ U(A) M(k) \backslash G(A)$. See for example the discussion after Lemma 3.3 in Gelbart-Jacquet "Forms of GL(2) from the analytic view point" for $G=GL(2)$ . I am sure somewhere in Moeglin-Waldspurger a similar lemma is quoted/proved somewhere for the more general $G$. Arthur has something similar certainly for $G$ reductive, but I remember that M-W consider also more generally finite covers etc. The main concern of these lemmas is actually the absolute convergence, but also they also provide the non-triviality.

A suitable set of representatives can be given via the Bruhat decomposition. Pick a compact set $K \subset U(A) M(k) \backslash G(A)$ containing only one representative and having non-empty interior. Let $\varphi :U(A) M(k) \backslash G(A) \rightarrow \mathbb{C}$ be a function which is positive, smooth, compactly supported on this set $K$, non-vanishing on the interior. Then your sum will not be zero only for $\gamma^{-1} g$ for $ g \in K$. This works equally well if you work modulo the center.