well it's not legally binding, your contract will still have a minimum notice period of 2 weeks to 4 weeks... I think your problem will be that by the time you start you'll be handing your notice in a week later, if I was your employer I'd fire you if I knew you'd been messing me around so you might find yourself jobless anyway
you definitely won't get a reference but tbh it's onyl a month of work anyway, it won't particularly improve your CV and won't damage you to just leave it off