Problem facility in district's centre (6)
I believe the answer is:
teaser
'problem' is the definition.
(I know that teaser is a type of problem)
'facility in district's centre' is the wordplay.
'facility' becomes 'ease' (I've seen this in another clue).
'in' means one lot of letters goes inside another.
'centre' says to take the centre.
The middle letters of 'district' are 'tr'.
'ease' put into 'tr' is 'TEASER'.
(Other definitions for teaser that I've seen before include "One who irritates playfully" , "Hard question" , "Difficult problem concerning Easter" , "Introduction to arouse curiosity" , "Easter (anag) - puzzle" .)