I also had trouble.
If you can craft the bowgun and/or better armor, that might help--unfortunately I don't recall where I picked up the schematics/parts. Other than that, apples heal a bit more than berries, and play around with cooking--there are some foods that slightly increase your speed, attack, and defense, and last all day. So you can eat them in the morning and still carry a full stack of healing items.
For the automated turret, the key is to stay in the shadows or behind cover. Use your dash ability to get between shadows, and once you start getting close to the turret, pay attention to the timing. If it can see you (green targeting reticle), it fires three times and then pauses--so get in a location where it can target you but there's something blocking its fire, let it shoot the obstacle three times, and then you have a few seconds to push the block before getting back behind cover again.