I have a mobility clause in my contract. I look after a 10 year old girl in the morning and at night so my family do need me.

Questions regarding whether a person breached a term of a contract are nearly impossible to answer without actually reviewing the contract itself. This is so because every contract is worded differently, and even slight differences in wording can make a huge difference. You should take your contract to an attorney to review. Also, note that breaching a contract does not necessarily make the contract void. That means that, even if you wind up in breach, the employer can forgive the breach, if they chose to do so. In other words, you might be able to work something out with them. Again, this is something you should discuss with an attorney.

Having reviewed the additional information you provided, it would appear that the employer has the right to have you move locations for a week. Regardless, it is not clear what the effect of you refusing to go might be. A contract must be reviewed as a whole document for it to be fully understood. Individual clauses, although informative, do not paint the whole picture. You should take the entire contract to an attorney to discuss your situation in more detail.