harassment/breach of confidentiality - can i report this clinic to the ico?

If your partner is listed as a next of kin or emergency contact, or they contacted you on a landline at the property which your partner has access to, and you gave them that information...then it's not really a breach of confidentiality is it...? I doubt they would disclose what happens within the appointments or even what the appointments are for without your consent as that would be a breach - but trying to contact the patient (including via next of kin or other listed numbers/addresses if they do not reply) following non-attendance, without disclosing the reason for the appointment, is probably reasonable.
I also wouldn't consider this harassment, and following up on DNAs even if you were the one paying for the appointments is probably good practice I imagine.
Also if your partner is the one paying for it then they probabyl have a responsibility (possibly even a legal obligation) to inform them of appointments missed because they probably will be charging for those and need to notify the card holder of any changes to the charges as a result.
Note that healthcare providers are legally required to retain certain health records for a minimum length of time: https://www.bma.org.uk/advice-and-support/ethics/confidentiality-and-health-records/retention-of-health-records so I doubt you will be able to have them delete your data.
If you don't want them to contact your partner you can tell them that and even specify that you can only be contacted by certain channels due to whatever issues you have with them contacting your partner. That said again, if your partner is the one paying they will have to contact them about any payment related issues as they would be the cardholder for any payments made (or pending).

yeah, he's listed as a next of kin/emergency contact but surely that should only be used in emergency situations? which wasn't the case here. they are just mad because they can't profit from me anymore

what about right to erasure? it states that you can request deletion of your date if it's no longer necessary - which it isn't, because i no longer want to be their patient. do you there's a chance my request will be approved or am i wasting my time?

Because the law states it is required to be retained i believe that probably qualifies as a necessary reason for them. I suspect you won't be able to do anything although you can always ask them - they may hold other data on you also that they can remove if you request it as well anyway.
As for the NOK contact again particularly as he is the bill payer I dont think its unreasonable to contact to query DNAs which probably incur a charge.