at around 40 knots which is below a safe flying speed.
Well, because you have included the word "safe" in that statement, I'm going to agree with you in this context. However, it is certainly possible to have a 172 in sustained flight during takeoff at a slower speed, and fly it away. The challenge is that you can get into very high angles of attack doing it, and you are getting closer to the stall (which is just settling back in this condition), or banging the tail on the ground, which never good. Stop watching the ASI, look out the windshield, and feel what the airplane is telling you, the airspeed is only one part of the whole thing.
The stall warning is just that - a warning. If you are not planning to stall, then don't pull back any more. But, if you're in controlled flight, not settling, and accelerating, you do not need to lower the nose either. Acceleration takes time (or you'd have been pinned into your seat!). So give it time, safely controlling it, and it will fix itself. The super fast correction you're applying 'cause your instructor told you to is a form of over controlling. Yes, a pilot can cause a less good condition of flight instantaneously by control inputs which are poorly timed relative to what the aircraft is going to be doing in a few seconds, and tens of second hence. Patience......
If you are in the air at 40 knots, and returning to earth is inevitable, would you not much rather do so in a flared attitude? Keep the nose where you had it, a little "tail low" (to use Cessna's term). A 172 is amazingly shock absorbent in that attitude. If you contact three point, or worse, nosewheel first, the nose strut, firewall, and perhaps propeller, and engine damage is going be costly.
Yes, practice, but unfortunately you cannot practice uneven runway techniques on a smooth paved runway. You can practice just holding the aircraft attitude and getting airborne at a lower than normal speed, and that's excellent, but anything you do on a smooth runway, which seems to induce an "uneven" surface will get very bad fast. If you're going to practice "tail low" takeoffs in a 172, do so with as forward a C of G as you can accomplish. In a lightly loaded 172M or later, I would expect to be happily in ground effect, accelerating, with maybe just a brief peep of the stall horn, at 40 knots.