Is "auto-relight" (or whatever Airbus calls it) basic on all A320s? It's basic on the 777, 787, and 747-8, but not on some of the other engine/aircraft combinations (on the 787, 747-8, and some 777s they even got rid of continuous ignition - auto-relight was certified as being just as effective). As an example, auto-relight was basic on the 747-400/CF6, but an option on PW4000.
Auto-relight will normally restart an engine due to a simple flameout, although it's not 100%. At least on Boeing, cycling the fuel switch will cause the FADEC to 'reset' (on rare occasion, the FADEC can get the equivalent of the a PC 'blue screen of death' - a simple reset will clear it and it can once again control the engine). Cycling the fuel switch should also cause the FADEC to switch channel in control - sometimes there are undetected faults in the channel controlling the engine - cycling the fuel switch gives the other channel a chance even if the first channel still thinks it's healthy.)
I've worked FADEC for 25+ years - they are smart, but not that smart (the very first generation of FADEC only had the equivalent of 24k of memory
). On a twin I'd suggest a relight attempt if there is no evidence of damage.