Determined person not getting on is needing shelter (10)
'determined' is the definition.
(I've seen this before)
'person not getting on is needing shelter' is the wordplay.
'not getting' is a deletion indicator.
'needing' says to put letters next to each other.
'shelter' becomes 'tent' (tent is a kind of shelter).
'person' with 'on' removed is 'pers'.
(Other definitions for persistent that I've seen before include "Unrelenting - prettiness (anag)", "Ongoing", "Endless", "Determined", "Resolute - interminable".)