None of them are correct in my opinion. You have here just, so it's the sign to use present perfect. That can be escaped, but it's better to use it. The final part talks about the future as it's the consequence of the receiving of the photos that you're now posses. The final version from me looks like this: I hope that the photo I've just sent you will make you feel better.

Number 2 is best if you correct the verb from make to makes, the photo... makes... Both work, but can make is very unusual. Using it can after saying I hope makes it sound like you hope it has the possibility to make someone feel better, not that you hope it will definitely make them feel better. That is why saying can is not totally right. Order of preference 1) ... makes you feel better. 2) ... will make you feel better. 3- with different meaning) ...can make you feel better.

#1 is correct as it is written. #2 add an "s" to "make". "I hope the photo I just sent you makes you feel better."