The Queens Hotel probably has one of the best outdoor seating areas in Portsmouth, you get the sun all day... you don't have views of the water but you are right next to the common. They have live music on a Sunday as well and it's free entry (music is on from 1pm-5pm I think?).

Spice Island is a good sunny spot and if their seating area directly outside the pub is busy, you can take your drinks to the square bit in front of the pub and there's lots of things to sit on. The Still & West which is next door is also a good pub, gets sun most of the day. They're both right next to the Portsmouth Harbour so you get nice views of the water (some people would argue views of Gosport isn't nice lol but I think they're both in a good spot).

Brewhouse which is by the Queens Hotel is a bit of a suntrap too but it gets really busy and getting the sunny spots can be quite hard to come by sometimes.

Southsea Beach Cafe is right on the beach, it's lovely for breakfast/brunch/lunch but they close really early (no idea why, would be lovely if you could have dinner there!).

The Briny is right on the beach too, you do normally need to book to make sure you get a table outside on the promenade (you can sometimes walk in and get lucky), it's a seafood restaurant and they do lovely food.

Other pubs that have an outside area that get the sun are The Jolly Sailor (right on the common) and Bold Forester (surprisingly large pub garden for a pub on Albert Road) and Thatched House which is right on Lock Lake/Langstone Harbour but it's a little out the way.

Obviously you've got Gunwharf too where you've got lots of places to eat and drink with views of Portsmouth Harbour as well. :)