When do we say that a given clock could be further corrected? In a recent report on experiments with a sample of ultracold ${\rm {}^{87} Sr}$ atoms, T. Bothwell et al. (physics.atom-ph:2109.12238), the abstract ends and culminates in the following punchline:

"This heralds a new regime of clock operation necessitating intra-sample corrections for gravitational perturbations."

(Unfortunately, the suggested "corrections" are not further specified, or even only mentioned, anywhere in the article; and I haven't had the opportunity to check the extensive list of references for clues. Has some particular method of "correcting atomic clocks for gravitational perturbations" been explicitly described elsewhere already ?)
However, I also have a general question on the purpose of applying a "correction"; or in other words, on deciding whether (and in how far) certain evaluations constitute "a correction" to a given clock:
Considering that it has been argued, for instance here, that "a clock is supposed to measure the arclength of its path through spacetime", is likewise (or perhaps even more strictly) a corrected clock supposed to measure (ratios of) arc lengths of its path segments more correctly than an uncorrected clock (on the same spacetime path) ?
 A: The measurements shown in this paper are made by shining a tightly focused laser beam onto a cloud of Sr atoms, where the cloud has a vertical size of about 1 mm. This allows comparing the frequency of the clock transition between different parts of the cloud. Figure 3 shows a plot of the measured frequency shift over the vertical extent of the atom sample. From this, it can be seen that after subtracting all known errors from effects other than gravity, a finite slope remains: the clock frequency depends on position. Hence, measuring the average clock frequency in the entire atomic cloud (for example, to get better signal-to-noise ratio) would yield a transition shape broadened by the gradient of the gravitational field.
Simply said: the clock's size is finite, therefore its top follows a different path through spacetime than its bottom. You could fix this by:

*

*Making the the clock smaller. This will be difficult at some point, because using fewer atoms or higher atomic densities will cause other errors to grow.

*Moving the clock into weaker gravitational fields. Probably there are already people working on this, but bringing an atomic lattice clock so far away from earth is certainly expensive.

*Calculating the expected differential shift and compensate it in data processing. This latter method seems to be what the abstract is pointing to.

