I see a couple things going on here with your question. First to answer the headline question:
The further away from your body you have your hands, the more you involve your chest. The trade-off is more stress on the shoulders and pectoral tendons.
This bit of advice also works for bench press. To further activate your chest, concentrate on tensing the pectoral muscles while you are at the top for a second each rep.
Dealing with Tendinitis
Tendinitis is a warning that either there is something wrong with your form, or that you are not exercising the antagonist muscles enough. For example, if you are doing lots of push ups which train your triceps, but not doing any type of rows/pull ups/flexed arm hangs to strengthen your biceps. If you are sticking with body weight exercises, you can do body weight rows by laying on the ground and pulling yourself up to either a bar or a table. You could even use a broom handle across two chairs.
By your description, you may be dealing with both a form problem and unbalanced training. Balancing your training is easy. Just add the antagonist work to what you are doing. Initially, you will want to do more antagonist work than the push ups, but eventually you'll want to do equal volume on them. The form is more challenging.
- Perform planks to teach your body to remain rigid. Lay on your stomach and get up on your elbows. Tighten up your entire body, in particular your abdominal muscles. You should be able to hold that position for at least 30 seconds at a time. If not, work on these until you can.
- Find a comfortable hand position that is slightly away from your body, but not so far as to cause discomfort in your shoulders. The position should help you engage your pectorals and your arm muscles while keeping the body plank rigid. Slight elbow flare is acceptable. Just make sure you are well supported.
- Only perform good reps. If you feel your body starting to fatigue and your form start falling apart, stop the set. You can work on adding more good push ups over time.