'Drop foot' .... laminae 'falling out'??
Never heard that one before! The blue will be the copper you're using.
Your horse has 'seedy toe', aka 'white line disease'. It happens when compromised hoof tissue allows bugs from the ground(can be bacterial or fungal) to get in & start eating away at the wall. It appears there may be another section on that side up at the toe that is also starting to separate. With good management it can generally be easily treated.
Firstly the cause of the 'compromised' hoof... no. 1 you say your horse is in muddy conditions. This is unfortunately great for softening horn & also growing bugs. If it's the only option for dry footing, I think stalling the horse for part of the day, to give the feet some chance to dry out is probably for the best. No. 2 it appears the foot has quite long heels & contracted, thrushy looking frog. It also appears that the toe is 'stretched'. Mechanics & compromised hoof function is a major factor in separated hoof walls & allowing the bugs to get a grip. Nutrition & diet also play a big part in hoof health, including susceptibility to infections.
Copper sulphate is indeed one thing that is commonly used/effective for seedy toe. Nothing short of heavy chemicals(which I prefer to avoid if poss) tends to work reliably in every case, as there are different bugs & other factors that different substances react differently to. Eucalyptus or t-tree oil, sulphur, peroxide(weak formula) plain salt, iodine, even, believe it or not, raw honey... are among other (gentler) effective measures.
As well as addressing the causes and topical treatment, unless perhaps the horse is in otherwise ideal environment/hoof form, etc, I also consider it is usually necessary to open up & clean out(known as resecting) the infected horn with a hoof knife first. Sometimes this is hardly more than scraping the infection out if it's shallow, but often the infection may go up a long way & a substantial section of horn may need cutting out. It is very often impossible to treat effectively just from the outside unless it's rather superficial. Your farrier should be able to do that for you.