I bought an iPhone 4 from a youngster who had put his own Apple ID on it but could not remember the password. I backed it up yesterday and restored to my new iPhone and it copied the Apple ID across too. Now I have 2 phones with it on. Anything I do?

that must be really annoying !!

so this is what you call

FIND MY IPHONE ACTIVATION LOCK Find My iPhone Activation Lock - Apple Support

when you start using iCloud on any iOS device and enable find my iPhone, this prevents a new user from using it, and also prevents anyone from using the phone

as it is locked to previous owners account ,this is also the case if when a phone is lost or stolen.

the only way around this is to have the previous owner reset their apple id by going to @ applied.apple.com

if they are unable to do this, then the only other way around this is to have the previous owner get a copy of their original receipt and call apple support who can then lift this lock if they get a copy of the receipt

unfortunately there is no quick fix -> the below link explains what you should do before you sell or give away an iPhone

What to do before selling or giving away your iPhone, iPad, or iPod touch - Apple Support