*This is an answer to the edited question*

Consider the short exact sequence
$$
0 \to \mathscr I_X(l) \to \mathscr O_{\mathbb P^k}(l) \to \mathscr O_X(l) \to 0,
$$
and the long exact sequence of cohomology it induces:
$$
0 \to H^0(\mathbb P^k, \mathscr I_X(l)) \to H^0(\mathbb P^k,\mathscr O_{\mathbb P^k}(l)) \to H^0(X,\mathscr O_X(l)) \to H^1(\mathbb P^k, \mathscr I_X(l)) \to\dots .
$$

By Serre's theorem $H^1(\mathbb P^k, \mathscr I_X(l))=0$ for $l\gg 0$ and hence the previous map $H^0(\mathbb P^k,\mathscr O_{\mathbb P^k}(l)) \to H^0(X,\mathscr O_X(l))$ is surjective.

You will not be able to get injectivity. In fact, the larger the $l$, the "less" injective that map is. $\dim H^0(\mathbb P^k,\mathscr O_{\mathbb P^k}(l))$ grows at the order of $l^k$ while $\dim H^0(X,\mathscr O_X(l))$ grows at the order of $l^{\dim X}$, so there is nothing you can do to guarantee injectivity. Quite the opposite, the only time you might have injectivity is for small $l$'s.