Subtle click on left Shimano 4700 STI brake lever when applying brake while moving forward
Sounds like it could be one of several things:


Loose headset. This is the most probable as it's not uncommon to need to re-adjust the headset as the bike breaks in a little, and it also fits with it being the front brake that's causing this and doing it a little bit with any amount of motion, but not otherwise. You can google "loose threadless headset" and get all the info you need about the topic, but the fast way of determining that it's the headset versus anything else (pad play in this case) is standing beside the bike, locking up the front brake lever with one hand, putting your other hand's fingers right over the gap between the stationary and rotating parts of the headset, and using your body weight to rock the bike forward and back, feeling for any undesirable movement at the gap. Check both the headset uppers and lowers. Lifting the front wheel off the ground a couple inches and dropping it is also a pretty good test on a rigid-forked bike that's not too junked up with accessories, although if there's any clunking you have to then differentiate whether it's from the headset or front hub.
Play from the front pads moving in the caliper. Most or all disc brakes have some degree of this, but usually the amount you can feel while riding is on the very subtle side, and it's also not usually an acquired problem, i.e. it stays the same for a given brake or set of pads. If you determine that this is what it is, there's nothing you can really do about it.
Theoretically, loose caliper or rotor bolts. Easy to check, and you'd be getting brake rub.