Kansas City and St. Louis each tax 1% of earnings, self-employment profit included, on a return of their own, separate from Missouri's. Live in the city and it taxes all of your profit, wherever you earned it. Live outside it and it taxes the share earned there - here, by the gross each gig paid, the way Kansas City's Schedule Y allocates by receipts. Log where each gig was.
A sole proprietor files Kansas City's RD-108 (a nonresident allocates on Schedule Y; it is filed even in a loss year) and St. Louis's E-234 as type E-2. Calendar-year returns are due April 15. An extension buys time to file, not to pay - Kansas City's RD-111, St. Louis's E-8 with 100% of the tax. Put in the year's net profit from your Schedule C and Soulard works out what each city is owed.