A lot of times prior to teaching students how to balance in half moon, I have them practice activating their feet. (Take a look at foot exercises!)
The reason for this is to create a stable foundation.
In a yoga pose like half moon, because the foundation is so small it can be very helpful to make it as stable as possible. That means learning to activate your foot and ankle. In half moon this can be a little bit challenging because the supporting foot is turned outwards relative to the pelvis.
However, you can counter this limitation by being aware of your hip joints and your pelvis.
Especially when first learning to balance on one foot, I'd suggest shifting your weight towards the front of your supporting foot so that your toes will naturally press down. In addition make the foot and ankle "strong" so that it is easier to stay balanced.
With the hand on the floor it can be easy to relax the foot. If you wish to lift the hand, I'd suggest stabilizing the foot first. Press the toes down and restablize foot and ankle, then lift your hand. When standing up from the pose, try to keep your weight towards the front of your foot so that your toes continue to press down even as you return to upright.
First turn your right foot out 90 degrees.
Shift weight onto your right foot and as you do so activate the foot and ankle. Shift far enough to the right so that your toes press down. Then lift your left foot.
To begin with lift the left foot an inch (or a few centimeters) and hold while staying balanced.
Turn your pelvis so that it faces the front. Then, to lift your left leg higher focus on lifting the left side of your pelvis.
If you find that your leg is wobbly despite your feet and ankles being strong, focus on keeping your knee and shin pointing to the side (in the same direction as your foot) and push your inner thigh forwards (in the same direction as your foot is pointing.)
Do this while keeping your foot stable.
Try to roll the left side of your pelvis backwards so that your pelvis faces the front. To do this try using the outer hip muscles of your right (supporting) leg. Then try to lift your left leg higher. Initially try to get it horizontal and then see if your can lift it higher by using the outer hip muscles of your left leg.
To touch your hand to the floor you may find it helps to bend your knee a little. You may also find it helpful, at least initially to use a yoga block.
Once you touch your hand to the floor you can work at gradually straightening your knee.
If using a yoga block, see if you can gradually reduce the height of the block (turn it) till you can touch your hand to the floor.
Once your hand is on the floor and you are in half moon yoga pose you can practice shifting your weight.
You can repeat the exercises as follows.
After a few repetitions see if you can stand slowly and smoothly, in full control.
Note that this pose can be a little more challenging if you look up as opposed to down.
Another exercise is to try closing your eyes. I like to close the eyes for a second (focusing on feeling the foot) the open. Then I repeat, gradually extending the length of time I keep the eyes closed.
Another challenge is to bend the elbow of the supporting arm and try to move the side of the ribcage towards the standing thigh.
Learn how to use Friction to improve leg and arm strength.
Simple exercises with easy to follow instructions
Making difficult poses like Chaturanga Dandasana easier to learn.
Learn Your Body with
Frictional Arm and Leg Strength
PDF or Video