Bulk-boundary　correspondence　(BBC)　remains　the　central　topic　in　modern　condensed　matter　physics　and　has　received　a　boost　of　interest　with　the　recent　discovery　of　non-Hermitian　skin　effects.　However,　there　still　exist　profound　features　of　BBC　that　are　beyond　the　existing　framework.　Here,　we　report　the　unexpected　behavior　of　BBC　when　the　Hamiltonian　contains　terms　of　the　form　d　0　(　k　)　I　,　which　serves　as　a　momentum　dependent　energy　shift.　For　Hermitian　cases,　the　momentum　dependent　energy　shift　can　force　the　system　to　be　metallic,　where　topological　phase　transitions　can　take　place　with　the　upper　and　the　lower　bands　kept　untouched.　The　modified　BBC　should　be　reconstructed　from　the　perspective　of　the　indirect　band　gap.　In　non-Hermitian　cases,　skin　effects　are　found　to　be　capable　of　coexisting　with　the　preserved　BBC,　of　which　the　process　can　be　greatly　facilitated　by　the　complex　d　0　(　k　)　I　.　Remarkably,　such　results　can　be　led　to　a　further　step,　and　contrary　to　the　intuition,　the　modified　BBC　in　Hermitian　systems　can　be　restored　to　be　conventional　by　including　extra　non-Hermiticity.　The　physical　origin　for　these　phenomena　lies　in　that　d　0　(　k　)　I　can　drastically　change　the　point　gap　topology.　Finally,　the　corresponding　experimental　simulations　are　proposed　via　the　platforms　of　electric　circuits.　©　2024　IOP　Publishing　Ltd