As teachers and as "at home teachers" we often think our job is to tell the student what the next step is in refining a skill or preparing a piece. That is an important part of our job, of course, but kids experience every instruction or constructive criticism the same way. No matter how we present it, kids usually hear "You're doing this wrong and now we have to fix it." (I highly recommend the book Helping Parents Practice, by Edmund Sprunger, in which he explains with great clarity how children experience practice psychologically). So it should be no surprise that kids who only get that kind of interaction from a teacher or parent can lose confidence and motivation.
Of course there are times when we simply have to address a problem, or introduce the next step, but the most effective teaching moments are usually not when we say, "okay, now do D-E-F," but when we say "wow! you did A-B-C so well." When children receive sincere, specific praise for a task, they are motivated to continue working as hard - sometimes even harder - to perform that task well. The same is true for adults. Think of the employers you've had. Were you more motivated to work hard for the ones who offered a lot of praise or for the ones who only ever pointed out where you had room for improvement?
When practicing with your child at home, most of your comments should sound something like "Great, Lizzie! You just did such a good job of keeping your bow on the highway, and it made your cello sound big and beautiful." Even if you're frustrated that Lizzie still isn't keeping her left elbow up after weeks of working on it, you can still always find room to offer honest praise.
Often if a parent complains to me that practice has really been a struggle, and that there have been a lot of arguments at home, I'll suggest that the parent only offer praise for what the child does successfully, and no 'instruction' of any kind for a whole week. Inevitably, both the parent and the child come back the following week feeling much better about things.
Furthermore, it is incredibly important that we only address one thing at a time. Sometimes it's very hard not to bombard a student with seventeen different instructions at once, but that only serves to overwhelm the student. Best to identify the issue that is most in need of work, and focus on that (see the post about Dr. Suzuki's Teaching/Practicing Priorities). If there are other things to work on, write them down and come back to them the next day.
Sometimes a child just needs to be allowed to explore and try things and figure things out. The parent should just sit back and watch and congratulate the child when he arrives at the correct solution (or a good solution) on his own.
Finally, sometimes the most helpful thing you can say to a child is "I love you."