As practitioner, it does frustrate me when people say 'I didn't know I had to ask permission'. I do understand that new tenants can be so desperate for keys that they'll take a tenancy on any terms offered but that doesn't stop them reading their tenancy agreement or handbook about what they can and can't do? To prevent this very situation, wouldn't a quick phone call to check first have been worth the effort? I agree it is your home, but it's not your property. You won't be there forever and if your children succeed to your home, will they be prepared to foot the bill for its removal? Planning permission aside, if, as you say, the structure is not overlooking or causing a problem then common sense and reasonableness could come into play in letting it stay as long as the right agreements were in place for its maintenance and removal at end of tenancy.
As for the rent arrears question, I'm assuming if they've started steps for repossession, then a revised Notice Seeking Possession will include ALL current grounds for possession as best practice suggests there should only be one live NSP - so if you had already received an NSP for rent arrears then a new one for breach of tenancy (improvements without permission) could prejudice the earlier one. Bringing them all together on one just continues their right to commence action on any of the grounds covered if required.
Another ominous sign here is that there appears to have been no face to face discussion - always in my book a sure fire way to create a stand-off.
In any event now that you have applied for retrospective planning permission, the HAs actions might be immaterial as the Local Authority Planning Enforcement team might seek its removal if your application is unsuccessful.