No sure I can advise any better Sally, what with the SA type behaviour we're getting from Pip at the moment - although we have the opposite problem, in that he's fine being left in his crate or kitchen while we're in, but screams the place down when we're out. 
I would be tempted to just remove the gates if you're never closing them anyway, but as you say, it is advisable to have some way of leaving him separate to your dcs when you can't be in the same room as them, so I can see your dilemma.
I followed the to get Pip happy to be left in his crate while we're in, as he was frantic before that and actually got his jaws stuck around two bars of the crate in panic.
We also 'captured calmness' by saying 'settle - good boy' every time he was lying quietly, now we can say it when we need him to lie quietly for a while and he has a conditioned response to it. If we say 'settle' now he heads into his crate and lies down.
Another idea would be to sit just the other side of the gate and chuck a treat in every time he stops digging/complaining, even for half a second, gradually building up the time you wait before treating so that he's quieter for longer and longer.
Sorry to hear about your Mum, by the way, hope she's making a good recovery. I'm sure Henry will be great company for her when she can finally come home.
Still haven't come up with anything Pip can safely chew while we're out - looks like we're going to have to try pigs ears then.