The answer is yes (as in the case of Sasha's answer we use ramified covers)
Proof. Let $X$ be any variety in $\mathbb CP^n$. Take a section $s_m$ of $O(m)$ on $X$ such that $s_m$ is not equal to $m$-th tensor power $s^{\otimes m}$ of any section $s$ of $O(1)$ restricted on $X$. Now, let $s_m^{\frac{1}{m}}$ be the multi-section of $O(1)$ on $X$. This multi-section defines a subvarity $\tilde X$ X_m$ in the total space of $O(1)$ on $X$, that is the cover of $X$ of degree $m$.
Finally notice that there is a family of maps from the total space of $O(1)$ on $X$ to $\mathbb CP^n$ that sends the zero section of $O(1)$ on $X$ to $X$. The image of such a map in $\mathbb CP^n$ is just the union of all lines in $\mathbb CP^n$ that join a fixed point $p$ with all points of $X$, the point $p$ itself does not belong to the image. Then the image of $\tilde X$ X_m$ in $\mathbb CP^n$ is the desired variety. END.
We used here the fact that $O(1)$ on $\mathbb CP^n$ can be embedded in $T\mathbb CP^n$ as a subsheaf (in various ways).
