Let $F$ be a compact oriented surface with a foliation $\cal F$ with $k$-prong singularities only (or, if it helps, assume that $\cal F$ admits an invariant measure). Is it true then there exists a pseudo-Anosov $\phi$ such that for every loop or an arc (with endpoints in $\partial F$) $\gamma$, $\phi^n(\gamma)$ is transversal to $\cal F$ for large $n$?

Here is a rough argument why it is probably true: The $MCG(F)$ action on ${\cal PMF}(F)$ is minimal, cf. Fathi-Laudenbach-Poenaru, Thurston's Work on Surfaces, for closed $F$. (A reference for bounded $F$ would be useful although the original proof seems to work). This implies that the unstable foliations $\cal F^u_\phi$ of pseudo-Anosovs $\phi$ are dense in ${\cal PMF}(F)$. So it is reasonable to expect that there is $\phi$ with all angles between $\cal F^u_\phi$ and $\cal F$ greater than some $\alpha>0.$ (But I don't have a rigorous argument for that.) Now by the work of Thurston (cf. the book above), the angles between $\phi^n(\gamma)$ and $\cal F^u_\phi$ converge to $0$ as $n\to \infty.$ Consequently, $\phi^n(\gamma)$ is transverse to $\cal F$ for large $n$.

EDIT: To address the comment below, let us just assume that $\cal F$ admits an invariant measure of full support.