The following is an execise from Chris Wendl's book Holomorphic Curves in Low Dimensions
Let $(M,\omega)$ be a symplectic manifold and $A\subseteq M$ a closed subset. Let $J_A$ be an $\omega$-compatible ($\omega$-tame) almost complex structure defined on a neighbourhood of $A$. Show that $M$ admits an $\omega$-compatible ($\omega$-tame) $J$ which restricts to $J_A$ on a neighbourhood of $A$.
There was a hint: Choose any $\omega$-compatible ($\omega$-tame) almost complex structure $J'$ defined on $M$ and a smooth homotopy between $J'|$ and $J_A$ on some neighbourhood of $A$. Then use a cutoff function.
Can anyone offer some help towards a solution of the exercise?
I'm comfortable with the fact that the spaces of $\omega$-compatible/tame almost complex structures on $(M,\omega)$ are non-empty and contractible, so I see the existence of $J'$ and the homotopy in the hint, but I don't understand how to use a cutoff function.