I have a fork suitable for short drop brakes and I have Shimano tourney brifters (ST-A070) with the gear cable coming out of the side of the hoods and the brake cable going under the bar tape. Which brakes are compatible/ would you recommend?
Specific product recommendations are off topic here, but general guidance can be given.
The Shimano ST-A070 shift/brake units are for a road drop bar bike, and I assume 'fork suitable for short drop brakes' means you need standard road bike rim calipers .
Generally, all standard road bike rim calipers work with drop bar levers. The levers pull the correct amount of cable for the leverage the calipers provide.
The obvious answer would be Tourney calipers from the A070 range, but these appear not to exist. The next level up (Claris 2400 series) does include road calipers though. I believe Tektro and Promax make some inexpensive road calipers as well.