= Solution
A <lift> of $f:Y\to X$ is a continuous map $\widetilde f:Y\to\widetilde X$ satisfying $p\widetilde f=f$. The <homotopy lifting theorem for a covering map>[covering homotopy theorem] says that a <homotopy> $H:Y\times I\to X$ has a unique <lift> once a continuous <lift> of its initial map $H(-,0)$ is prescribed. No additional local connectivity assumption on $Y$ is needed for this <homotopy>-lifting statement.
To prove injectivity on fundamental groups, let a loop $\widetilde\gamma$ based at $\widetilde b$ project to a based null-homotopic loop. <Lift> its null-<homotopy>, starting with $\widetilde\gamma$. The two endpoint tracks <lift> constant paths at $b$ starting at $\widetilde b$, so remain at $\widetilde b$ by uniqueness of path lifting. The final path lies in the discrete fibre over $b$ and starts at $\widetilde b$, so it is constant. Thus the lifted <homotopy> contracts $\widetilde\gamma$ relative its basepoint, proving
$$
\boxed{p_*:\pi_1(\widetilde X,\widetilde b)\hookrightarrow\pi_1(X,b)}.
$$
In the following constructions, take $x$ around the <circle> factor and $y$ around the projective-plane factor. Above each sheet point, the <circle> has one incoming and one outgoing lifted $x$ edge. A fixed sheet for $y$ carries an identity cover of $\mathbb{RP}^2$; a transposed pair of sheets is joined as the antipodal fibre of a <sphere> covering $\mathbb{RP}^2$. These pieces give genuine covering neighbourhoods at the wedge point as well as away from it.
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