Riding on the trail for one day a week won't cut it for your horse to be able to mostly trot and canter a 5-10 mile hunter pace course, if you want to be competitive. You should to strive to ride as often as possible, every ride builds up your horses stamina and muscles, but be careful to not over do it. For every day of training, your horse gains fitness, but for every day off your horse looses those days of training. Think of it as a bank account and if you put in 7 days of riding and take 7 days off your horse is back to zero. However, riding 7 days a week isn’t recommended and will be too much on your horse. Every other day, or about 3-4x a week is a good number to strive for. For example, ride Monday, Wednesday, Friday, then give your horse Saturday and Sunday off (plus Tues, and Thursdays off). Another thing you can do is on days where your not riding you can lunge your horse, or instead of riding, 20-30 minutes of lunging in the ring is beneficial. This is also a solution if you are pressed for time on a given day. Walking and trotting/cantering in the ring is a good idea on and off, but 20 mins on and off will get very boring for you and your horse, especially 3-4x per week. Add some jumps, and use things like ground poles. You are still conditioning but you are also teaching other things to your horse like balance and stride. Try to be outside if you can, (I know, I hate mud too!) but just walking outside will keep your horse interested, even if it's just down the road (providing it's safe!). A challenging trail ride as one of the sessions is also a good idea, and gets your horse out of the ring and onto terrain that you are likely to encounter during the hunter pace. Also, if you can, go to where the hunter pace is going to be held and get a feel for the terrain and try to mimic that in your horse’s training. I also recommend if you can, check out the course a day or two before to see how the terrain is holding up. However, I don’t know if previewing the course is permitted, depending on how serious the particular event or course is.
You may also want to look at your own fitness. If after you ride, the next day you are sore and with time it doesn’t seem to get any better, you may want to start your own exercise routine. Walking or jogging/ running on days you don’t ride are beneficial, as well as some small weight training. I like to walk at least 3 days a week, or run and it builds leg muscle, and I just feel better in every aspect of my life.
Please keep in mind that I am not a professional, but I too am starting to condition my horse (and myself!) for hunter paces and this is close to what I am doing, so I hope it helps.
Also-Sorry if you’re a conditioned/more advanced rider and if some of this seems like a no brainer, but I just figured I put a base out there for people to think about and build on no matter what level they are.