Two variations on side plank:
-when in position, dip your lower hip down towards the ground (so you make a v type shape) and then lift back to straight position, repeat as many times as you can manage, aim for three sets of 15-20.
- when in position, raise your upper arm straight so pointing skywards, with palm forward. Then bring it down in front of you and then through the gap underneath you. This tips your body round slightly, working your waist. You can also do this holding a small dumbbell if you want to make it harder.
Both of these can be made (much
) harder by supporting your body on a hand, with straight arm, rather than resting on elbow, same as you can make a full plank harder by using press up position arms.
To make a full plank harder rest one foot on top of the other, or use a bosu ball/Swiss ball to rest your feet. Basically anything which makes it harder to balance - a smaller point of contact such as one foot, or an unstable surface, will make your core work harder.
If you go to the gym, bent over rows with a barbell are great for strengthening your back, just be careful with weight you use with your existing problem. Try maybe 2.5kg on each end and see how you get on, work up if ok.
If you are willing/able to throw cash at this, one of the best pieces of equipment for core and back exercises (and pretty much any other body part) is a TRX suspension trainer, very useful to have at home and to take with you if you travel. You can get one for around £100 on Amazon....lots of vids on YouTube of people using them. I have one and nothing touches it for core strengthening!
HTH x