Hi Chris
This is going to be a tricky search as you are working with such a big time line.
You can only really go by the time the last child was born ~ to the Census where his wife says she was widowed.
Now, as I'm sure you are aware, while we would all like to think that the Census are completely accurate they often aren't, people lied, people fibbed, people made guesses, people made errors.
Then there is the Census taker themselves, they too were open to human error OR making presumptions.
So, you have a date when the last child was born.
Just a scenario, but he could have abandoned his wife and children.
His wife, on the following two Census when asked, says Married, to cover embarrassment and of course to a degree she still would be . . .
. . . then there is the Census where she states she was widowed . . .
. . . But, just for one second, presuming that he did abandon his family . . . wherever he went to, he could have died and his wife wouldn't have known . . .
Was there then a presumption by the latter Census, in her own mind that he was dead, he may not have been . . . so you still need to look for him AFTER this date.
So, again, I'm not saying that this is fact, just a possibility . . .
. . . but you could be buying a LOT of certs to try and work it out, wait on certs 'til you get a much better lead.
IF it were me, I would look at every single Michael Henry with at least a 5 year age variant in all of England, Scotland, Wales (at a later point then the US)
Start at the first census after the last child was born . . . start with one of the Michael's that you will find.
My instinct is that his occupation won't change so stay looking for painters.
Be flexible with places of birth as we still can't be 100% sure that Sligo or indeed Ireland is correct, so be open to any POB.
In Ireland, people in prison or institutions sometimes appear by just their initials, I'm not sure if the same applies in the UK but try searches this way too.
Don't rule out a possible match that may have a wife and children, worse case scenario he may have remarried.
A lot to think about and take it but a bit of lateral thinking may yield results.
Tara