Why would Lord Hornwood want his bastard son raised at Deepwood Motte and why would House Glover accept him? Do you think Laurence Snow’s mother from House Glover?

Fostering a bastard child with another house is a pretty normal way of handling the situation with minimal embarrassment – Edric Storm being fostered with Ser Cournay Penrose, etc. Lord Hornwood minimizes tensions with his wife and her Manderly cousins, and Lord Glover enjoys stronger ties to House Hornwood (who now owe them a favor). 

And there’s potential advantages down the road, as we see when the Hornwood Question arises. With the legitimate male line struck down in battle, all of the sudden the lands of House Hornwood are up for grabs – both the Manderlys and the Umbers want to maintain the widow’s usage rights as a way to bootstrap their way into a claim, the Tallharts who have a claim through the female line want to put forward their younger son as the nearest male heir, etc. If Robett Glover were to marry his daughter to a legitimized Laurence, that gives House Glover a strong claim indeed to the Hornwood lands.