I’ll start out by saying you sound worn out and wrung out@Redcar87 . It’s true that young children are Hard work at times, however much we love them, and sometimes it feels a bit relentless- but I think you are struggling and definitely need to seek more support in some form or another. You do seem down, possibly depressed, so I think you should make an appointment with your GP and also talk to your husband about extra support. If his mental health isn’t in a place where he can do all of what is needed, I think you may need to speak to family, or even friends, if you can.
All of that said, whatever the reasons, I don’t think you are heading in a good direction as parents and you and your husband (as it’s not all down to you) need to take a long hard look at things and try to implement changes before it gets harder to retrieve the situation. The longer this goes on, the more difficult it is to fix.
Specifically, I think you are creating a rod for your own back by letting your children do things they shouldn’t in order to avoid a tantrum. It will just get more difficult and you will find it harder and harder to manage them. And it isn’t helping your children, in the short-or long-term. We all let the occasional things slide, of course we do. We all let our kids eat something they would probably be better off not eating (though I’ve never yet needed to allow chocolate for breakfast) or let them watch tv for longer than we normally would to get some peace and quiet. But these should be rare occasions, not the norm.
If you are letting your sons misbehave/ damage or waste things/ upset others etc rather than step in to correct them because you can’t face dealing with a tantrum; or allowing them eat junk food, treats and snacks instead of meals, or whenever they ask or just keep them quiet for a bit; or allowing- probably evenly actively encouraging them (given their ages)- to spend long periods (? most of the day) plonked in front of the tv, rather than having meaningful interaction or play are regular/routine occurrences then you do need to make changes because it is poor parenting. Your children will lose out in the long run and you will not be happier for it as a family.
Children need boundaries, children need to be taught what is acceptable and what is not, they need their parents and the adults around them to show them right from wrong, how to behave in any given situation, to model good behaviour and relationships, to show them how to manage disagreements and conflict sensibly and so on. Children push boundaries (hence the tantrums)- this is absolutely normal behaviour. Having boundaries makes children feel safe, however, even when they rail against them. If there are few or no boundaries many children push further and further to find the limit, or some become very contained, controlled (to their detriment) and sometimes very anxious. There are differences in where those boundaries are drawn between parents, but they are needed (equally overly harsh/critical/rigid rules and boundaries aren’t healthy either, obviously).
If you don’t have rules, expectations of behaviour etc, then how will your children learn these things? If they are never corrected, never stopped from doing things they shouldn’t or made to do things they should/must do even if they don’t want to, they will never grow into rounded individuals. They may well struggle at school, in friendships and so on because they will not be able to do what they want, as they want, when they want or get their own way all the time in these environments, but are not equipped to deal with those situations.
Children need balanced meals- and I’m not one for saying no sweets/crisps/junk food ever, they are all fine in moderation- and to develop healthy eating habits. They will go through phases of being picky, some children are worse than others too- but sensible eating habits are important. Equally, too much TV in place of playing with other children/being outside/being read to/doing activities with parents and others is not good for physical or mental development and health. Using screen time for large chunks of the day to allow you to check out/get through the rest of the day isn’t healthy for them. Or you.
I don’t mean to be unkind or unfair. I say these things as a reality check, I suppose. It’s clear you do love your sons and care about how you are parenting them- I also think you know what’s happening isn’t ideal, or you wouldn’t have posted here. I think you are struggling with two young children and a partner with mental health problems, and need some help. You need to seek that help so that you can nip this pattern in the bud, feel better and be a better parent. Start with your GP or health visitor, and speak to your husband too. Perhaps also family (parent or sibling?) or a close friend.