We study the eigenvalue correlations of random Hermitian $n\times n$ matrices of the form $S=M+εH$, where $H$ is a GUE matrix, $ε>0$, and $M$ is a positive-definite Hermitian random matrix, independent of $H$, whose eigenvalue density is a polynomial ensemble. We show that there is a soft-to-hard edge transition in the microscopic behaviour of the eigenvalues of $S$ close to $0$ if $ε$ tends to $0$ together with $n\to +\infty$ at a critical speed, depending on the random matrix $M$. In a double scaling limit, we obtain a new family of limiting eigenvalue correlation kernels. We apply our general results to the cases where (i) $M$ is a Laguerre/Wishart random matrix, (ii) $M=G^*G$ with $G$ a product of Ginibre matrices, (iii) $M=T^*T$ with $T$ a product of truncations of Haar distributed unitary matrices, and (iv) the eigenvalues of $M$ follow a Muttalib-Borodin biorthogonal ensemble.