Hi Mark G.
Yes you can open an ISA with another provider and transfer the funds you hold in your account, once the transfer has been fully completed you can transfer it back to the first company if you wish to do so as long as you haven’t re-invested for this tax year. Although please bear in mind the company you are transferring to legally have 30 days to complete the transfer process so if you are going to do this it could take a few months before your money is sitting happy in an account of your choice.
As for adding money if doesn’t affect your interest in any way, as long as you hold a balance in the account even without reinvesting would still gain interest on any funds you hold in there – it is only 0.20% AER at the moment in time for existing customers and 2.60% AER for new applications.
If you wanted any further info call the customer services and ask to speak to banking enquiries who deal with ISA’s, the free number to speak to customer services is – 0113 2766166
I hope you find the best deal you possibly can get on your money and that I’ve helped in some way – if anything I’ve saved you a phone call maybe?