Hi, Jess! Welcome to the forum! Now, before I go on, let me see if I understand correctly. You have a female bird that is called Joe? Also, you got this bird from a trainer or a rescuer? And how do you know your bird is one year old? Does it have a closed leg band with 2015 engraved on it? Because, if it doesn't, then you don't really know how old it is and, hopefully, it's older than that. I say 'hopefully' because if your bird is indeed only one year old, it would be considered a juvenile and juveniles don't pluck unless they have a very serious medical problem. I also doubt it's a pied. There are pictures on the net of senegals that appear to be pied but, in my personal experience, the 'out of place' yellow feathers are a direct consequence of plucking and not a genetic mutation. Not that there could not be such a mutation because Lord knows it seems that parrot breeders are mostly concerned about getting weird mutations and nothing else...
Now, further to Wolf's posting on light. He talked about the light schedule (photoperiodism -and you can do more research on it, lots of info on the net, just put avian photoperiodism on the search engine and you'll see) but the quality and placement of the light source is also very important. I like good quality full spectrum lights in the ceiling fixture. I recommend Dr. Mercola's full spectrum bulb because it has a high CRI and the right Kelvin temperature (the right specifications would be CRI 94+ and Ktemp between 5000 and 5500). There are avian lamps that clamp on the top of the cage as well as avian bulbs for sale but I don't recommend them. For one thing the avian bulbs tend to have too high an UV output and putting a light so close to a bird can result in burning of their corneas (and rendering the bird blind in the process) or their skin. Always do your own research about bird stuff because the pet industry is completely unregulated and they can say something is good when, in reality, it can be real bad for them.
You might also want to evaluate its diet. The thing with senegals is that they tend to be one-person birds and can be VERY aggressive toward anybody/birdy they perceive as competition for their chosen one's affection - and this aggression is intensified with a bad light schedule (which Wolf already covered) and a high protein diet. Their aggression is also what prompts me to ask you to tell your children not to overwhelm the bird (I would actually tell you to tell them to ignore it but I had children and now have grandchildren and now it would be futile to ask them to do that

). Joe is now on what we call the honeymoon period and that means the best behavior you will ever see from it. This is the period of time that the bird is super nice and analyzing the situation and new people so this is the actual foundation of what his relationship to anybody in the household will be and, when it comes to a senegal and children, you want to do everything exactly right. I used to have a rescue and would take in aggressive birds and of all the mean birds I've had (and I am talking macaws, ekkies, amazons, conures, lovebirds, etc), the worst was, without a doubt, a Senegal. Both of mine were given up because of aggression, the male toward mankind in general and the female against the owner's wife and his other birds until the wife gave him an ultimatum, her or the bird. Senegals are not naturally sweet-tempered birds, they are intensely jealous and completely fearless and a child would not stand a chance against an aggressive sennie, they might be small but their bites are real bad (I have one on each arm right now, courtesy of the male Senegal which is a bit peeved because the female is not 'putting out').
I agree 100% with Wolf about the beak and wing trimming - only when there is medical reason and only done by a professional (groomers are not formally trained for this and, besides, they are service providers so they will always tell you their services are needed whether this is true or not). I worry about Joe losing its balance twice... Unless the nails and wings were trimmed super duper short and the perches are completely inadequate (which I doubt), this should not have happened. Babies are a bit clumsy but at one year +, the bird is no longer a baby so there is some other reason for this instability. Was this bird never vetted by the person who rescued it and later gave it to you? Because, usually, birds only fall from a perch when they are having a seizure or they have neurological problems (their feet actually lock in the perching position if they squat a bit -that is what enables them to sleep without falling while standing on a perch) so, if it hasn't been vetted, if I were you, I would take it to an avian vet asap and not wait a couple of months. Better safe than sorry, right?