In　this　paper,　the　pth　moment　exponential　stability　is　investigated　for　impulsive　stochastic　time-delay　systems　(STDS)　with　random　impulsive　delay.　A　novel　concept　of　average　random　delay　in　impulses　is　introduced　to　deal　with　the　pth　moment　exponential　stability　of　STDS,　which　provides　a　new　idea　for　the　study　of　the　stochastic　delay　in　impulses　from　a　holistic　perspective.　Considering　the　impulsive　effect,　the　destabilizing　and　stabilizing　delayed　impulses　are　studied,　respectively.　It　is　revealed　that　the　random　delay　in　impulses　plays　a　significant　role　on　the　stability　of　the　system,　which　may　not　only　destabilize　a　stable　system　but　also　stabilize　an　unstable　system.　In　particular,　our　results　allow　the　delays　in　continuous　dynamics　and　impulsive　dynamics　to　exceed　the　length　of　the　impulsive　interval,　which　are　more　general　than　previous　results.　Finally,　some　numerical　simulations　are　presented　to　illustrate　our　proposed　results.　©　2023　Elsevier　Ltd