Sign of hesitation entering museum gallery (7)
I believe the answer is:
veranda
'gallery' is the definition.
(synonyms)
'sign of hesitation entering museum' is the wordplay.
'sign of hesitation' becomes 'er'.
'entering' means one lot of letters goes inside another.
'museum' becomes 'V and A' (Victoria and Albert Museum).
'er' placed inside 'vanda' is 'VERANDA'.
(Other definitions for veranda that I've seen before include "Roofed platform along the outside of a house" , "Roofed terrace along the side of a house" , "Sheltered terrace around a house" , "Stoop" , "balcony" .)