Suppose a set $X\subseteq H(\omega_1)$ of hereditarily countable sets is parameter-free definable in the structure $(H(\omega_1),\in)$ by a formula in the language of set theory, say $\phi(v)$, i.e. $$(H(\omega_1),\in)\models\phi[x]\text{ iff }x\in X$$ for all $x\in H(\omega_1)$.
Can we then define the set $X\cap H(\omega)$, i.e. the subset of all hereditarily finite sets in $X$ in the structure $(H(\omega),\in)$ of hereditarily finite sets, i.e. is there a $\psi$ such that $$(H(\omega),\in)\models\psi[x]\text{ iff }x\in X\cap H(\omega)$$ for all $x\in H(\omega)$?
No. For example, the set of (Gödel numbers of) sentences true in $(H(\omega),\in)$ is definable in $(H(\omega_1),\in)$ but not in $H((\omega),\in)$.