I don't think the black-and-white thinking about poverty is helpful. Yes there are common themes but there are so many different factors at play.
Some people end up in poverty through a change in circumstance. If this is permanent disability or long-term ill health it can be permanently life changing but otherwise this is the group most likely to be able to pull themselves out of poverty given time. They'll be the ones with the well-adapted strategies to coping that can help them manage the situation. However, they can get trapped as well. If, for example you have pets and a car your benefit money will disappear in no time as it was never designed to include these sorts of costs. But getting rid of the pets can be heartbreaking (particularly when they are an emotional anchor, especially for children) and getting rid of the car might mean isolating yourself from job opportunities. So you carry on paying for them. The money gets tighter. Then the washing machine breaks. It gets put on a credit card because at this stage you still have access to credit. But now you have a debt to service too, so your money goes less far again. Then the kids need new shoes, so the credit card balance creeps up more than you've been able to pay it down. The debt needs more servicing and the benefits don't cover everything. You give up the car to save insurance costs, petrol etc but now you have to spend money on buses, shop more locally and more expensively and there are fewer job opportunities. The small amount of money you got back for the car quickly runs out topping up these extra costs, clearing the credit card, replacing a few household items long overdue, buying the kids their (secondhand) winter coats and new clothes because they've outgrown everything. And now they need new shoes again and it's Christmas/one of their birthdays. You have £10 to feed the 3 of you for a week. Iceland is the only shop you don't have to spend money to get to. They are offering BOGOF on pizza, nuggets and chips. You can get 4 or 5 meals for £5. Buy the cheapest loaf of bread for £1, cheap marge and some cheap ham and yoghurts for packed lunches. Over £8 now. Need some milk too. Pennies left. You have no flour, tinned tomatoes, herbs or oil in the cupboard to consider buying veg/pasta and cooking from scratch. £3 on spices, £1 flour, £1 oil before you add any actual ingredients, think about packed lunches or milk...
Benefits are set at a rate you can live on reasonably well in the short term only and only if you live in a well-appointed urban area with good public transport. The minute an unexpected cost crops up things can spiral very quickly indeed. And that's assuming no one has any additional needs which further push up costs.
Then we have people who may never have been taught or experienced some of the skills some people seem to think are innate. Or who may be so used to spending all their mental energy trying to feel safe, managing the behaviour of those around them, managing their own issues etc that there's nothing left over for planning food prep or a long term strategy for getting out of poverty.
Everyone's story is unique and in my professional and personal capacity I've seen a lot of them. What I do know is that very few people 'feed their kids crap' because they're lazy. There is usually an explanation if you care to look for it rather than write people off.