I think it's important to note that when people hear about the 20k cap, they think that means people are routinely receiving 20k in benefits. They're not. It also sidesteps the number of people who need to be supported.
20k a year for a family of four is very different to 20k a year for a family of six or seven. The only reason people end up with this level of benefit, generally speaking, is because Child Tax Credits are paid on a per-child basis. So we are talking about capping benefits for large families here.
This leads to "if you can't afford them, don't have them!" but without getting into the ethical or practical arguments around this - it's too late. We're talking about kids who exist already.
That's who this will punish.
The Tax Credits system already ensures that - on a person-per-person basis - a household will have more income if someone is working. Broadly, a family of four on 20k will be better off financially than a family of five on 22k. So there is already a mechanism in place to incentivise working.
So ultimately, all this does is ensure that children will grow up in greater poverty. This is socially counterproductive as study after study shows that if we want to counteract the long term negative effects of poverty on a person's life (including worklessness, lack of skills, not paying tax), early intervention is the thing that works best and most consistently. Counteracting poverty in early years by ensuring that children have secure, warm homes, decent food, access to stable education and nurturing environments (and yes, abuse rises when financial stress rises), is absolutely key.
The benefit cap may make many current adults feel better. It will not, in any way, address the "benefit culture" of worklessness those same adults are so concerned about in upcoming generations.
It's the social equivalent of cutting off our nose to spite our face.
As to the suggestion that benefits be capped at the same rate as a full time minimum wage job? Someone working a full time minimum wage job with a child would be eligible for full Child Tax Credit and up to 70% funding of childcare on top of that. This is because the government recognises that currently, it is possible to work a full-time job and STILL not have enough money to feed and clothe a child. Being in receipt of a significant amount of benefit can have literally nothing to do with how hard you work, or how much you work.
Since there will always be jobs that are the lowest paid, and since we will always need people to fill those jobs, I think it's unreasonable to demand that the cost of holding one of those positions is childlessness. I would be perfectly happy if the government chose to address this by raising the minimum wage instead, but so far, that's not the option they've chosen.