In　the　continuous-time　random　walk　model,　the　time-fractional　operator　usually　expresses　an　infinite　waiting　time　probability　density.　Different　from　that　usual　setting,　this　work　considers　the　tempered　time-fractional　operator,　which　reflects　a　finite　waiting　time　probability　density.　Firstly,　we　analyse　the　solution　of　a　tempered　benchmark　problem,　which　shows　a　weak　singularity　near　the　initial　time.　The　L1　scheme　on　graded　mesh　and　the　weighted　shifted　Grünwald–Letnikov　formula　with　correction　terms　are　adapted　to　deal　with　the　non-smooth　solution,　in　which　we　compare　these　two　methods　systematically　in　terms　of　the　convergence　and　consumed　CPU　time.　Furthermore,　a　fast　calculation　for　the　time　tempered　Caputo　fractional　derivative　is　developed　based　on　a　sum-of-exponentials　approximation,　which　significantly　reduces　the　running　time.　Moreover,　the　tempered　operator　is　applied　to　the　Bloch　equation　in　nuclear　magnetic　resonance　and　a　two-layered　problem　with　composite　material　exhibiting　distinct　memory　effects,　for　which　both　the　analytical　(or　semi-analytical)　and　numerical　solutions　are　derived　using　transform　techniques　and　finite　difference　methods.　Data-fitting　results　verify　that　the　tempered　time-fractional　model　is　much　effective　to　describe　the　MRI　data.　An　important　finding　is　that,　compared　with　the　fractional　index,　the　tempered　operator　parameter　could　further　accelerate　the　diffusion.　The　tempered　model　with　two　parameters　α　and　ρ　are　more　flexible,　which　can　avoid　choosing　a　too　small　fractional　index　leading　to　low　regularity　and　strong　heterogeneity.　©　2022,　The　Author(s).