$R$ is artinian algebra over field $k$(not necessarily commutative), $M$ is $R$-mod with $pd_R(M)=1$ where $pd_R(M)$ denotes projective dimension of $M$ over $R$. Then projective cover $P\to M\to 0$ kernel contains one of primitive idempotent. It is clear that kernel of the map $K$ is projective by $pd_R(M)=1$.
$\textbf{Q:}$ If $K$ is f.g. over $R$, then it follows from classification of f.g. projective modules over $R$ that $K=\oplus_i(Re_i)$ with $e_i$ primitive orthogonal idempotents of $R$. Now note that $M$ is any $R-$module. In particular, $M$ could be non-finitely generated. So $P$ could be non-finitely generated. Hence there is no guarantee $K$ is finitely generated.(One can't apply classification theorem of f.g. projective modules over artinian algebra over the field $k$.) It is not clear that $K$ contains a copy of principal ideal $Re_i$ of $R$ for some $i$. Why $K$ contains a principal ideal?
$\textbf{Q':}$ If $K$ contains a principal ideal, why does $P$ contains a projective submodule containing $Re_i$?