"Which support has good mobility and has good HP?"
The unfortunate news is: Tankyness, actual high-mobility, and being good at supporting aren't qualities that are really found together.
The best advice I can give is this:
Try some out yourself so you don't waste your hard earned money or your harder-earned IP.
If you play some you can and find out which supports have a playstyle that works with you, that you find engaging and enjoyable, and is something that you will find fun worth your money.
You can't be helped with that by asking on a forum because people are just going to tell you what they enjoy and that's not what you enjoy.
Try the free-rotation!
Right now there is one support on the rotation and he has a lot of effective HP.
Tahm Kench - the River King.
He is a tank Champion with the ability to regenerate the health that he loses from damage or can instead transform the damage he has taken into a shield that prevent him from taking anymore. He can support as well because he also engulf a target allowing him to take a target to safety or to chase down a fleeing enemy, or devour an enemy champion for a single target nuke that scales even without items.
Based on what you said, you might like him. Watch some videos on how to use him and give him a shot.
Alistar is another option and you can get him if you subscribe to Riot's youtube channel.
He can charge and Headbutt enemies away, let out a roar to regenerate his health and those around him and can ult to take 50-70% reduced damage from enemies. He is another option for your tanky support needs.
I'm giving you these as no-risk options, because I know it would suck to spend a months worth of IP on a champion you did not like and would not play based of a board rec.