Revised version of previous post - with new graph

Let me try and provide some answers, although provisional, and the measurements could do with being repeated.
Glenn is perfectly correct - complex question with many strands.

Ulli - advantages:
First, removing gross frequency errors likely biggest improvement for many for WSPR as well as FST4W. Frequency errors of tens of Hz have not been uncommon - and so stations near one or other band edge will have been missed. 

Second, using the frequency comparison dashboard (or other method), differences between receivers and GPS receivers at the same site can be identified. I'm seeing several examples, including here in Southampton - so could lead to optimised GPS reception.

Third, as Glenn says, spectral spreading can be much lower than when there was uncontrolled drift in most installations other than Tom WA2TP. I say 'can' because even with this fix there are periods of poor corrections I can see here perhaps from poor GPS. What does that mean for your question on longer modes? From the diagram I did on a spectral spread 'budget' in the report at
http://wsprdaemon.org/ewExternalFiles/FST4W_on_HF_bands_V1-3.pdf
I expect no change in spot numbers for WSPR or FST4W-120 - those modes have a lot of headroom. But, for FST4W-300 the narrower KiwiSDR spectral spread could, as an approximation increase the probability of detection for two-hop propagation from about 50% to about 90% - probability falls off very quickly with spectral spread. But, it would remain unusual to decode FST4W-300 global paths with their greater spreading.

Tom - a provisional answer on how much better the Bodnar is is in the following (revised) plot.

These are histograms of spectral spread (as probability, because they all have different number of points; probability normalises them).
I am confident of the three KiwiSDR curves that are using their standard GPS aiding before v1.557 with browser as you were using, and after v1.557 without a browser connection.

I am confident that when a Kiwi has a Bodnar external clock (orange) it is much narrower. But it is so narrow that it was beyond my ability to measure correctly - I was measuring the spread of my reference oscillator.  I have very high confidence in the new data here - it is from Glenn N6GN's remote Kiwi with GPSDO with line of sight to WWV as the signal. It is the spike in brown. That's the jitter you can expect from your BOdnar  

For comparison in cyan is the QRP Labs QDX on receive I am confident that it is that narrow as the signal was the Bodnar.

regards
Gwyn G3ZIL

This is a complicated question with several pieces, propagation and band not least. I have no short answer but from what we've seen so far, there will be cases where the improvement will probably provide spots that would not otherwise have occurred, maybe even on WSPR2.

With the new code the Kiwi is SO much better!

How much will this software update improve the decoding quality of the new longer transmission modes - especially when NOT using phased locked GPS modifications - any expectations ?

Ulli, ON5KQ

John Seamons has fixed this problem in release v1.557 of the KiwiSDR software.

I have check here, the fix in v1.557 has worked. Over two hours the frequency difference between two updated KiwiSDRs showed an average of 0.09 Hz and an rms difference of 0.14 Hz from 333 measurements. There is no longer a need for the workaround of keeping browser connections open to a Kiwi.

Gwyn G3ZIL

I have posted the following at https://forum.kiwisdr.com and hopefully there will be a fix in due course. In the meantime there is a workaround as set out below. I am grateful to Glenn, N6GN, and Tom, WA2TP for checking out this problem.

As part of investigations into the use of FST4W on the HF bands, in conjunction with WsprDaemon, we have come across an issue.

If kiwirecorder.py connections are the only ones to a KiwiSDR (tested with versions v1.498 and v1.554) then the GPS derived frequency corrections do not take place. Rather, the frequency drifts. The issue and a workaround are described in detail in the attached pdf.

Immediately on opening a browser connection to the Kiwi - it can be admin or a receiver channel - the GPS corrections resume. GPS corrections continue so long as the browser window is active. If it suspends then the corrections do not happen until it is reloaded.

This provides a workaround - keep a browser connection open to the Kiwi.

With this knowledge, we are reevaluating our previous measurements of spectral spread as they may have been influenced by unnoticed drift.

regards

Gwyn G3ZIL