And "recurring" is the key word, since the theme is "Born To Rerun", Toobworld's version of reincarnation.
From the BBC's 'Doctor Who' website:
"Adric, a teenage boy with a gift - and badge - for mathematical excellence, was a member of a street gang on the planet Alzarius, along with his brother, Varsh. Adric, alone and hungry for adventure, stowed away in the TARDIS after Varsh sacrificed his life to save the Doctor and his friends.
Often frustrated by his clumsiness, and fiercely competitive, Adric struggled to find a useful role among the TARDIS crew. He ultimately gave up his life by remaining on board a spaceship that crashed into prehistoric Earth, causing the death of the dinosaurs."
Alzarius was found in E-space, a dimension separate from Toobworld's known space like so many of the alternate versions of the TV Universe. Whether or not the souls of its inhabitants can transfer to other dimensions to be reincarnated is unknown by this dabblin' sciolist. But Adric sacrificed his life in the dimension of Earth Prime-Time, and so it's pozz'ble, just pozz'ble, that he could be reborn as other TV characters in the main TV Universe.
For alls I know, as Stuart Best would say on 'Murphy Brown', since Adric perished at the end of the dinosaur age, he could have been reborn as animals or as representatives of the earlier stages of Man on the evolutionary scale.
I think that the basic nature of Adric's soul would remain the same He would be reborn as a delicate naif struggling to fit in with those around him, and his vaunting intelligence would sometimes be more of a hindrance than a help in that endeavor.
And that would bring us to Dr. Zack Addy. From Adric to Addy - it even sounds pre-destined!
So, as it is with most "Born To Rerun" suggestions, as well as with the "Theories Of Relateeveety", this can't be proven, but it's highly unlikely it's ever going to be disproved either.....