You have no right to a refund.
The right to a refund, in respect of faulty goods, only exists to the point when you're deemed to have legally 'accepted' them. This period isn't defined in law but normally only exists just long enough for you to get the goods home and try them out. Typically, this might be a day or two. By allowing 28 days, Carphone Warehouse are being extremely generous.
After you've legally accepted the goods, the retailer is only bound to repair or replace goods which fail due to an 'inherent fault'. (e.g. poor components or poor quality manufacturing). The customer can indicate which of these two options he would prefer but the retailer can reject the proposal (and substitute the alternative remedy) on the grounds of economic viability.
If the repairs to your phone have been ineffective, you can probably now demand a replacement phone (unless there is a reasonable probability of Carphone Warehouse being able to arrange a proper repair promptly). However, you still have no right to a refund.
The law might not be in your favour but the threat of bad publicity often works. Try mentioning the name of your local newspaper or the BBC programme 'Watchdog'.
Chris