Week by week tenancy

Tenancy Types and Management

Hi I have a tenant who should be vacating this evening. However he’s heart broken as his potential house has fell through . I have a new tenant viewing on Saturday and to be honest my present tenant has been poor at paying ; however he is almost up to date . He’s asking to stay as he will be homeless and he has his two children quite a lot .
I was pleased that he put in his notice as I was getting quite fed up of chasing rent etc from him. But I feel sorry for him and want to know if I put him on a week by week contract can I get him out if he defaults ? Ie can he be on constant agreed notice weekly ?