Does a sudden drop of the velocity of the headwind affect the airspeed by lowering it?
Your airspeed does not remain constant because of inertia: it takes more time for the airplane to adapt to the new relative wind, compared to the time it takes for the wind to change.

Example One: you're flying 80 knots and the headwind is 20 knots. Over a time of 3 minutes, the headwind gradually reduces from 20 knots to 10 knots. Since the change is gradual, you will notice the ground speed smoothly increases from 60 knots to 70 knots.

Example Two: you're flying 80 knots and the headwind is 20 knots. In 2 seconds, the wind drops from 20 knots to 10 knots. At the start (t = 0), you are moving 60 knots over the ground. However, 2 seconds later, the plane has not yet accelerated 10 knots faster (maybe only 61.5 knots), but the wind is now 10 knots. Therefore you experience an airspeed drop in 2 seconds from 80 knots to 71.5 knots.

Given sufficient altitude, the plane will eventually speed up to 70 knots ground speed (airspeed 80 knots again). When you are landing, you may end up touching down short of the runway if corrective actions are not taken in a timely manner.