By using angle-resolved photoemission spectroscopy (ARPES), the variation of the electronic structure of HfSe2 has been studied as a function of sodium intercalation. We observe how this drives a band splitting of the p-orbital valence bands and a simultaneous reduction of the indirect band gap by values of up to 400 and 280 meV, respectively. Our calculations indicate that such behavior is driven by the band deformation potential, which is a result of our observed strain induced by sodium intercalation. The applied uniaxial strain calculations based on density functional theory agree strongly with the experimental ARPES data. These findings should assist in studying the physical relationship between intercalation and strain, as well as for large-scale two-dimensional straintronics.