I'm divorced (against my will) and don't have children.
Here's my two cents, if anyone would like to hear. Here's what I would consider:
1) Am I willing to forego the shared experience of being a first time parent?
There's a lot of functional parental identity that married parents develop *together* through time and experience. It's is a severely different dynamic when one is an experienced parent with an established family identity, while the other has to jump straight into the deep end of the pool, rather than learning developing roles and being a team throughout.
Personally, this would be a MAJOR factor for me personally, as I really want to become a dad some day, but I really want to share that NEW experience with my wife. Sure, this is partly due to intimidation by having suddenly to become a parent to someone else's children, each with their own experiences, norms and expectations, and partly due to a sensitivity towards the feelings of the kids involved (having come from divorced parents) when uprooting their norms and setting up new ones, but mostly it's because I consider that shared experience of becoming a parent as a big part of developing a deeply-rooted marriage. I listed it first because I find it pretty insulting and unfair for it to be said a man isn't "Godly enough" or "manly enough" or even that he isn't "up to the challenge." That may not be true at all. Some guys just want to share the entire parenting experience with their wife and don't want to lose the bonds that form during that part of your lives, or the establishment of a unified parenting face to their children.
2) Why is she a single mom?
This is the biggest consideration. I understand sins and bad choices. I also know (as in my own situation) that divorce doesn't require the consent of both people. Most tragically, sometimes a spouse's life is cut short. While it's no good to condemn someone for their choices, understanding the decision-making of your potential spouse is necessary to gauge their character. Some might inappropriately call this judging, but if you won't gauge the character of the person you're courting, you can't determine if they have godly character. Are they a single mother because they decided to forsake their covenant, or because they were unwilling to sacrifice as needed to keep it? I find these to be very personal, but very necessary questions. I believe that if they are off-limits, there is likely an issue with an unrepentant heart that would be a huge red flag - too likely to be ignored. If she owns her mistakes, is she willing to be reconciled to her husband? If not, why not? Does she have issues with forgiveness/bitterness? Does it put me in a bind spiritually, marrying and sleeping with another man's wife just because things were too tough with her husband? I won't go there. But it begs the question - what is her perspective on the significance of and what is required of a covenant?
3) Is there another man raising and making decisions regarding my children?
This is the second biggest question for me. I don't want another man in my marriage/family. If the kids' biological dad is making choices for and raising "my" kids, that's not okay with me. ESPECIALLY if he is the sort that it was so important to get away from that someone broke covenant over. If it's okay enough for him to be around, it's okay enough to do the biblical thing and be reconciled. Either way, if there is another dad involved, it's just not okay with me. I have compassion for her, but I have more compassion for the children and not having to have multiple "allegiances" with multiple parents. I can fully understand a parent wanting to have a "godly" influence in the form of a loving new spouse, but from the kids' perspective, if the dad is still in their life, they already HAVE a dad, and don't want another - this only confuses things. They usually just wish their dad was there and and that their parents would love each other and be together. I also think it's terrible for kids to see their mother being with someone who they know is NOT their dad. For me as a Christian kid, it was gut-wrenching and heartbreaking. I don't think any kid should be faced with that. If dad is in their life, Dad should be dad. This is just my perspective. All this said, I had a step-dad who I REALLY butted heads with early on, but I really appreciate him as a man now. He's a good guy, but no kid should be placed in that situation when they already have a dad (again, my opinion).
I have a friend who married a gal with two bratty kids who she loves very much, and though she won't allow him to discipline the other man's kids (*facepalm*) he is the most patient, loving dad and step-dad a family could ever hope for and they have one new kiddo together. And I do believe he's an extreme blessing to the other kids and to his wife. I know God has and will continue to bless him. There are many instances of Godly men stepping up to be a parent and husband to wayward families. Just be as careful not to "judge" Godly guys for having reservations before leaping into a relationship with a single mom.
All of these things being said, after having my own kids, I'd gladly adopt kids at any age as part of the love/family ministry. I would love to show these kids unconditional love they'd given up hope for. My heart feels the same compassion for a single mom who feels no hope for having a godly Christ-following husband. To be clear, although I've studied it in depth, I'm not clear on God's heart regarding how to handle all of these modern situations yet. There are several excuses for divorce that we've culturally inserted recently that were not mentioned in scripture and were not accepted in that time. Frankly, Jesus revealed that even the human excuse in Mosaic law (infidelity) is not really God's will and is no excuse. And yet He applies mercy and grace for a repentant heart and keeps his covenant with us no matter what. I don't know what God's answer is for this, but I do know that He is loving, compassionate, and faithful. Make Him the center of all that you do and how you think, and He will provide over and above all that you need. And when things are tough, hold on and don't give up. All seasons are only seasons. It's not a hopeless situation, but Godly care should be taken, I think. Just my opinion.
Blessings in Jesus,
PolarGuyInAK