The Libertines guitarist's show at Norwich UEA on February 18 and a second at Bristol's Anson Rooms on February 21 have also been cancelled.
New dates have yet to be confirmed for shows in London, Portsmouth, Birmingham and Glasgow.
"Unfortunately, the upcoming UK tour has had to be postponed," a post read on Barat's official website. "Apologies for any inconvenience.
"Unfortunately, the Norwich and Bristol gigs have both had to be completely cancelled. Ticket refunds are available from point of sale."
The updated tour dates are as follows.
- March 29 - Brighton, Concorde 2
- March 31 - Cambridge, Junction
- April 3 - Oxford, O2 Academy
- April 6 - Manchester, Academy 2
- April 8 - Sheffield, Leadmill
- TBC - London - Shepherds Bush Empire, Portsmouth - Wedgwood Rooms, Birmingham - HMV Institute, Glasgow - ABC