This　paper　considers　a　two-dimensional　tempered　time–space　fractional　diffusion　equation　with　a　reaction　term　on　convex　domains.　Firstly,　the　analytical　solution　to　a　one-dimensional　tempered　time–space　diffusion　equation　is　derived　in　terms　of　the　Fox　H　function.　However,　for　a　two-dimensional　counterpart,　an　explicit　expression　for　its　analytical　solution　does　not　seem　tractable.　This　motivates　us　to　resort　to　numerical　methods.　Next,　the　L1　formula　on　a　graded　mesh　is　modified　to　approximate　the　Caputo　tempered　time-fractional　derivative.　A　fast　evaluation　for　the　tempered　time-fractional　operator　is　developed　based　on　a　sum-of-exponentials　approximation,　reducing　the　computational　work　and　storage　significantly.　Furthermore,　the　Galerkin　finite　element　method　based　on　an　unstructured　mesh　is　utilised　to　solve　the　problem.　Its　stability　and　convergence　are　established.　Finally,　two　numerical　examples　in　different　convex　domains　are　investigated　to　demonstrate　the　effectiveness　of　the　numerical　method.　As　an　application,　the　tempered　fractional　Bloch–Torrey　equation　retaining　Larmor　precession　in　a　human　brain-like　domain　is　illustrated　to　observe　the　evolution　of　the　transverse　magnetisation.　An　interesting　finding　is　that　the　tempered　parameter　has　a　significant　impact　on　the　decay　of　magnetisation.　©　2022　International　Association　for　Mathematics　and　Computers　in　Simulation　(IMACS)