In this study, inverse spectral problems for a energy-dependent Sturm--Liouville equations with $\delta$-interaction on a finite interval are considered. Some useful integral representations for the solutions of the considered equation have been derived and using these, properties of the spectral characteristics of the boundary value problem are investigated. The uniqueness theorems for the inverse problems of reconstruction of the boundary value problem from the Weyl function, from the spectral data, and from two spectra are proved.