Bobone, can't help you with the cancel, but it could very well be Galesburg-- don't live there anymore, but I'm from Galesburg, IL. BIG railroad town-- always heard that if he ever was able to invade the US, Galesburg was one of the 10 cities he targeted because to control there, he could paralyze the US Railroad industry....of course I can't quote the source of that, but heard it several times growing up.
To propose a scenario: If mailed short of the depot, the card would typically have a town cancel or a transfer clerk cancel if mailed at the depot, thus it appears the card was dropped in the slot of the side of the Galesburg (yes, Illinois) and Kansas City RPO mail car while stopped at a station or mailed by a passenger on the train. Breckenridge, MO is very close in line between Galesburg and KC, without doing the specifics on the actual rail route. The card would have continued east to Galesburg by rail where it no doubt continued by rail to Boston. Rail transit of mail as much as possible in this era. The Boston main PO clerks would have sorted it to the proper Boston station and the carrier took it out for delivery. There is no reason to have any en-route marks.
The RPO mark is listed at R=2 in the Mobile Post Office Society catalog, so reasonably common.
Breckenridge is relatively closer to KC (about 80 miles) and I think this was probably mailed at Breckenridge. If close to pickup time, cancelling of this would have been left to the RPO clerks anyway, who would also have to sort any mail from Breckenridge by destination. It was then carried to KC and onward.
The wrong way? The killer reads "W", frequently used with RPO cancels and indicating westbound, which jibes with this routing. But this then should be the quicker way of getting to Boston; there were express trains to the East from KC, likely including some carrying mail exclusively that likely might skip a very small town like Breckenridge. The only railroad serving Breckenridge was the Burlington as of 1896.
We don't know the exact routing from there as John Becker notes. The mail is already sorted and bagged in KC for its destination; no reason for marking any exchange unless something happened like a really major delay. The circular marking with "C" is of course a Boston receiving cancel.
I still believe the card was delivered direct to the train by the writer or mailed on the train, and here's why ... The Postal Laws & Regulations would not permit the postmaster at Breckinridge (or anywhere else) to send onward any uncanceled mail posted there. Postmarking was required under penalty of being reported as a delinquent postmaster (Sec 375 of 1879 PL&R). I see no exceptions - even in the paragraphs about mail received after the "closing of the mails". Unless someone knows of a regulation of that era I am unaware of.
West versus east. Yes, I saw the killer and I my initial thought was to take it with a grain of salt as being inaccurate, but on further thinking .... Is it overall faster to (apparently) backtrack to KC or to go through Galesburg and likely on to Chicago for a fast train east? I will waver for lack of exact 1884 knowledge. It seems like it would depend on the exact schedules of the day and the connecting trains. The postmasters and railway mail clerks of the day must have been amazing to know all this data. Scenarios seem logical of different mailing times during the day to dictate sending it in either direction due to making other connections, when does the next train came from a certain direction, etc. I will continue to waver. Regardless of initial direction, once it got on the first train, it would have stayed in the railway mail system until it got to Boston.
Skipped towns: Perhaps not stopping everywhere, but pickup through the use of a mail crane to catch the bag would equate to no skipping.