= Solution
Use the <homotopy lifting theorem for a covering map>: a <homotopy> of <paths> has a unique lift once its initial <path> is lifted. Also use <uniqueness of lifts to a covering space>: a lift of a constant <path> with a specified starting point is constant.
Let $\widetilde\gamma$ be a loop based at $\widetilde x_0$ whose image under $p_*$ is trivial. There is a <homotopy> $F:I\times I\to X$ from $p\widetilde\gamma$ to the constant loop at $x_0$, fixing both endpoints. Lift this <homotopy> with initial <path> $\widetilde\gamma$. Both vertical boundary <paths> are constant lifts at $\widetilde x_0$. Its final <path> is a lift of the constant loop and hence is also constant at $\widetilde x_0$. Thus the lifted <homotopy> contracts $\widetilde\gamma$ relative to its endpoints. The kernel of the induced <group homomorphism> is trivial, proving
$$
\boxed{p_*:\pi_1(\widetilde X,\widetilde x_0)\hookrightarrow\pi_1(X,x_0)}.
$$
No universal-cover assumption is needed for this injection; it holds for any actual <covering map>.
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