I'd like to offer some advice which hopefully will enable your child to continue with his playing. He has done very well up to this point. He is learning a specialist skill, it is hard. Learning how to play at home is also hard. It's a universal skill though, which can be applied to anything the student wishes to become strong at. It will greatly assist in his ability to independently learn. If the environment is right the student will develop their playing at home and at the same time begin to use some very important independence skills.
At home the student himself must take responsibility for their playing and not the parent. Reward and play are very important themes here. I remind students to reward themselves for putting the time in: something simple like making your favourite sandwich or something similar. I let students know that playing at home means your rewarded twice because the instrument, over time, sounds better,
The role of the parent is to allow the student to implement their own practice routine. Praising the child for doing the practice is a good thing. It reinforces the reward. Refrain from commenting on the quality of the music, unless it is something praiseworthy. In essence, the parent needs to step back and allow the student to have good weeks of practice and not so good weeks of practice. Do not over remind the student to practice. This can become negative and stressful. Instead ask how the practice routine is going?
Within the first few minutes of the lesson with me it is very clear how well the student has practiced. It is my role to encourage and motivate the student to reflect on their playing and to try and get it closer to the practice model I have showed them. I go over the 'new stuff' and review earlier material. I need to provide an atmosphere where the student feels OK to say how well their playing is going. I can then help them.
Please remember that it is better for the student to keep on going and have a few poor weeks of practice than to stop. The dream or desire to play is a long term thing. The skill of practicing, and the motivation to do it, take time to grow. They are not natural.