My wife has a small clock, of sentimental value only, which has a spring
with a weight on the bottom which oscillates vertically to operate the
escapement. It runs for a couple of seconds but the movement that the spring
exerts on the escapement quickly decays to a point that the escapement no
longer operates because the claws do not move far enough to release one
tooth at a time. I presume the escapement is not providing enough "driving
force" to keep the spring oscillating.

https://i.postimg.cc/50Kt1njT/20201013-131035.jpg

The spring has a peg trapped between its coils which causes arm 1 to move
clockwise/anticlockwise about a pivot as the spring oscillates. This causes
two claws to operate the escapement. There is a second arm, arm 2, but that
plays no part in the escapement - it just operates a moving bird on the
clock face!

The clock used to work perfectly, but has been in storage for several years
because we couldn't find a key to wind it until recently.

I can't see anything that looks obviously bent or out of place. The main
spring is fully wound. The coils of the "pendulum spring" are roughly evenly
spaced when the spring is extended - one coil of the spring is not more
stretched than another. The spring is securely attached at the top, so it
can't have rotated to move the peg onto the wrong turn of the spring, which
would cause the escapement claws to be unbalanced: it takes about the same
amount of movement of arm1 either side of its neutral position to release
either the upper or lower claw.

If the clock is placed face down so there is no load on the spring, the
spring can be manipulated so the escapement oscillates very rapidly and the
gears turn, which proves that the mainspring is driving the escapement and
there isn't excessive bearing friction.

Any suggestions?