Okay, here's an answer, although it might not be the one you're hoping for...
Having listened through to Stravinsky's Mavra following the score a couple of times, I can't find material significantly similar to Blanter's Katyusha either. That doesn't mean, of course, that there is no possibility of a link, but to my ears there is no easily discernible one.
Rather than simply assert that there is not likely to be a link, though, let me back this up with some reasoning and research. I'll do this in two ways: comparing the musical material in Katyusha and Mavra; assessing the reliability of internet sources that suggest a link between these works.
Comparing the music in these works:
I found quite a number of different examples of sheet music for Katyusha, none of which seemed definitive. (For instance, there is no "original" sheet music available for this song on IMSLP.) Although different examples vary a little, and some are in different keys, the example below shows the melody and harmony which best represents the various versions I found online. (See online examples of this song here, here and here.) I have notated the song in A Minor:
Internet sources suggesting a link between Katyusha and Mavra point specifically to Chanson Russe (1937), a later adaptation of Mavra by Stravinsky. However, Chanson Russe is not an adaptation of the entire opera; as Eric Walter White points out in Stravinsky: The Composer and his Works (2nd ed., 1979), Chanson Russe is an arrangement of just the Russian Maiden's Song from Mavra, firstly for violin and piano, then 'cello and piano. (Sheet music for the violin version can be found here.)
Indeed, if one compares the opening bars of Chanson Russe then Russian Maiden's Song, you'll see that the melody line and accompaniment are virtually unchanged (although, interestingly, the time-signatures are simplified). It is basically a transcription, rather than an arrangement:
Russian Maiden's Song from Mavra:
(Apologies for the poor quality of the pictures.)
Anyway, it is immaterial whether one is considering Chanson Russe or the Russian Maiden's Song from Mavra, neither seem to me to be similar to Katyusha. Although I have only shown the first few bars of these pieces, I can't hear similarities later in the pieces either. (I haven't shown the entire sheet music for Chanson Russe and Russian Maiden's Song here as they are copyrighted material, and besides they would take up a lot of space! I couldn't find sheet music for Mavra online, instead I got a copy of the vocal score from a London music library; as I mentioned earlier, music for Chanson Russe can be found here.)
In judging whether Katyusha is similar to the Stravinsky pieces, I have been focusing upon the melodic parts thus far; these are, after all, the most recognisable aspects of the musical material. If there were a significant link between these pieces, one would expect to hear them in the melodic parts.
However, there doesn't seem to be significant similarities in the harmonic progressions of these pieces either. The first section of the Blanter has clear blocks of tonic and dominant harmony, with the second section a little more adventurous, but emphasising the sub-dominant before arriving back at the tonic. The Stravinsky is quite different. There are no clearly defined "blocks" of harmony, instead the music alternates constantly between tonic and dominant 7th chords. Interestingly though (this is Stravinsky, after all!), the left hand outlines a tonic arpeggio throughout the first section, creating the impression of rather more complex harmonies when combined with the right hand. Also, the Stravinsky changes key twice for two phrases of the song, before returning to the home key for the last phrase. (This is even more obvious in the violin and piano version, as this version repeats the single "verse" of the opera version, giving the listener an extra chance to hear these changes of key.)
So, if not melodically or harmonically, how is Katyusha similar to the Stravinsky pieces? Well, possibly in "character" or "accompanimental style". But this is rather tenuous; similar characteristics could be found in a lot of music evoking "Russian folk song".
As I say above, internet sources suggest that Katyusha is based upon the Russian Maiden's Song from Mavra, but I can't hear a link. If instead, this is an error, and the link is with a different movement from Mavra, I can't hear this either.
Assessing the reliability of the internet sources suggesting a link:
Like you, I managed to find quite a number of online sources suggesting that Blanter's Katyusha is influenced by Stravinsky's Mavra. But I would question the reliability of these internet sources, for a number of reasons:
- most of the sources I found (eg. here, here and here - but also elsewhere) seem to have very similar wording. In particular, the Wikipedia pages in various languages for Katyusha use very similar wording. This suggests to me that many of these pages have simply copied information about this song from one original source (possibly one of the Wikipedia pages), without any further verification or research.
- each of these similar sources, including the non-English Wikipedia pages, use the phrase "some critics believe…" when asserting the link with Stravinsky. They do not say who these critics are or give references to where the original comments of these critics can be found. In other words, there is no actual evidence that any such critics exist, or if they do, any evidence of who they are and exactly what link with Stravinsky there might be.
- the English Wikipedia page about Katyusha does not mention any link with Stravinsky.
- the English Wikipedia page about Mavra makes no reference to either Blanter or Katyusha.
I could not find any online source, suggesting a link between these pieces, that either: gave a named source for someone suggesting such a link; explained what this link might be; provided a reference or link to such a source.
Having listened to all of Mavra, I can't hear musical material that could have influenced Katyusha, either. Personally, I wouldn't trust the various online sources that suggest such a link. None of them seem to provide a reference for an original source, and as the wording of many of them seems very similar, I suspect many of them seem to be simply copied verbatim from an original unverified source (eg. one of the Wikipedia pages).