This seems like a problem that should be easy if true, but I wasn't able to prove it in general or disprove it. Given a complex projective variety $X$ and a sub-variety $Y$ and a closed point $p$ on $Y$. The question is: Is it possible to move $Y$ so it does not include the point $p$?

By moving I mean the following: there is a continuous map $F: Y(\mathbb{C})\times [0,1] \rightarrow X(\mathbb{C})$, where $ F(-, 0)$ is the closed immersion of $Y$ in $X$. Furthermore the image of $F(-,1)$ does not include $p$. Moreover $F$ at any fixed time $t$ is an algebraic map ($F(-,t)$ is an algebraic map).

Another equivalent way of stating the problem is the following. Let's look at the connected component of the Hom scheme $\mathcal{H}om(Y,X)$ that contains the closed immersion of $Y$ into $X$. Is there another morphism in the same connected component that its image avoids $p$?

If untrue is there any general form for varieties $X$ that this holds?

I am interested in a very specific case where $X=\text{Sym}^n(\mathbb{P}^r_{\mathbb{C}})$. $Y$ is a subvariety that contains all diagonal elements i.e. points with multiplicity $n$ of the symmetric product. $p$ is one element in the diagonal of the symmetric product.