Was this match part of a tour? http://i62.tinypic.com/4j2wlg.png
RSSSF only has one match:
18/08/1938 St. Kitts & Nevis [a] w 4-2
Yes, I found only those two football matches on the tour, which seems to have been a multi-sport tour of Grenada to St Kitts.
17/08/38 - St.Kitts 0-4 Grenada
18/08/38 - St.Kitts 2-4 Grenada
But this is based entirely on the reports in the Gleaner, so we can't be sure that (in particular) the St Kitts teams were both proper representative teams (maybe one was A, one B or whatever).
It would not be a "St Kitts & Nevis" team anyway, I think. Just St Kitts.
Indeed because it was a multi-sport tour, beginning (as ever) with cricket, it is unclear if these were really the strongest football teams of the two islands, or perhaps they were sporting teams chosen mainly for cricket purposes, which also played at other sports? Table-tennis was also played on the tour, perhaps other sports too. The report of the table tennis match (see 22nd August edition of the Gleaner) says that the Grenada team left after that, and stopped at Nevis on August 20th. Unclear if any games were played at Nevis, or it was just a quick stop.
On one of our previous trips to the library, I wanted to look up the details of this tour in a local newspaper but there was no paper from St Kitts nor Grenada for that month at the British Library, nor at the Library of Congress. Infuriatingly the coverage stopped abruptly in July 1938. I might look at those anyway to see if there are any clues in the late July issues.