Child poverty has a lot of factors involved - as others said, there are a lot of structural issues, housing is a major factor. This may be a part, but I think it's more complicated than just more kids born to unmarried couples. I think that trend is largely because fewer people get married just because of a pregnancy -- and marriages that are based solely on a pregnancy, well, they're less likely to go well either even when statistically marriages are more secure than cohabitation. Some marriages are less secure than others.
There have always been people who have kids with little resources, that's not new, but we do need more resources to be at a 'normal' playing field in today's society. It's harder to get back up when you're down if the hurdle is really high - again housing costs are huge in this. Marriage has protective elements, but it can also lead to poverty if you marry the wrong person too that can be harder to dig out of and plenty of people in either are find to start with and then things go downhill. I think more resources for adults to retrain while working is important with how our work lives have shifted.
I do think education on marriage needs to include the legal structures around it. Part of me thinks that part of registering to marry should include being given as least a general rundown of the legal rights and responsibilities of marriage. This information is easily and freely available, but I think it should be treated more like the contract it is.
I like the idea of cohabiting for x about of time giving similar rights to marriage.
I disagree from an ethical standpoint - if I'm entering into a legally binding contract with another person and the state, I want to actively sign the contract, not have it happen as a passive default from the passing of time. I don't think the state should change my legal status without my active involvement.
I also disagree with a pragmatic standpoint - especially as I think adults shouldn't be discouraged from living together when it benefits them as that also benefits the housing stock. If adults are living together how, from outside the home, do you prove two adults living in the same house are in a relationship that could be counted as a defacto marriage? I live with 2 other adults and we've all lived together for over a decade now -- without the marriage certificate how do you prove from outside the home who is in a relationship with who and who is single? You can't just use my kids' birth certificates because things can change in the several years since then. We all pay different bills. There are cases where common law marriages is legal reality where people have had to prove they aren't in relationships, and it's very invasive and questionable. People who want to harm will use the law to their benefit so we have to balance giving them more tools to use with what protection such a law would actually give.
The law has changed since I got married to prevent immigrants as I was from marrying as I did. It was sold and is still discussed as a protective measure for vulnerable people, but I'm not sure it's actually protected anyone - I've seen no evidence of it. It's just been used deport people and attempt to reduce marriage visa stats (no real evidence it's done that either) by raising the age and adding more requirements which could have increased child poverty from parents having to be in separate countries. Now people can plan around that, but having been caught up in more than one immigration law change, some will do things based on law being X, and then it becomes Y, so we can't just say that people know what they're getting into when there will be people caught out as the laws transitions and it affects personal lives.