Let $d>1, n>1$ and consider the vector space $V=\mathbb{C}[x_0,\ldots,x_n]_d$. Let $X \subseteq \mathbb{P}(V)$ be the set of all forms with the property that their zero locus in $\mathbb{P}^n$ has at least $k$ singularities. Let $p \in \mathbb{P}(V)$ be square free such that its zero locus has less than $k$ singularities and all of them are nodes. Is it true that in that case $p$ is not in the Zariski closure of $X$?

Let $d>1, n>1$ and consider the vector space $V=\mathbb{C}[x_0,\ldots,x_n]_d$. Let $\mathscr A \subseteq \mathbb{P}(V)$ be the set of all forms with the property that their zero locus in $\mathbb{P}^n$ has at least $k$ singularities. Let $p \in \mathbb{P}(V)$ be square free such that its zero locus has less than $k$ singularities and all of them are nodes. Is it true that in that case $p$ is not in the Zariski closure of $\mathscr A$?