The incidents are said to have occurred in the Town Centre on Tuesday 5th May.
The first burglary occurred at around 02:00 at Market Bakery in Scarrots Lane. Substantial damage was caused to the inside of the premises, an untidy search was carried out and a cash register was stolen.
At around 03:50, The Retreat Café in Scarrots Lane was also broken into and food was removed from inside.
Police are appealing for witnesses who may have seen a man described as wearing jeans, dark trainers and a dark T-shirt, between 03:50 and 04:20 on 5th May in the vicinity of Scarrots Lane and South Street.
Although the image is not the clearest, officers hope someone will recognise this man.
Anyone with information is asked to contact police on 101, quoting 44200158083. Alternatively, you can contact the independent charity Crimestoppers 100% anonymously on 0800 555 111, or via their anonymous online form at crimestoppers-uk.org