I have 2 mechainc's wires:
One is 0.028" and stiffer than the larger one, which is 0.042" and dead soft. I use the smaller one to do those passages. I put a slight, gentle bend on the end of the smaller wire, about 1/8" or so from the end, to let me turn it this way and that to make it snake thru the hole. Then I push the debris out with multiple strokes toward the needle jet's hole in the center of the throat.
I suspect that the debris in mine came from having parked it in 2001 with MTBE-laced gas in it when I had the big cancer operation that kept me out of the saddle for almost 6 years (I couldn't manage the bike's weight for a while). When I started back up with it, I couldn't ride to work (4 miles) without having to replace the plugs when I got home: 3 of the 4 would be fouled black each time. I did this for a week before deciding it wasn't going to "clear itself out" and then it started snowing again (this was in March) so I took off the carbs to find out. I had never cleaned that passage before, but the carbs were sparkly clean [from sometime during the sojourn when I cleaned them], yet they fouled plugs immediately, with any gas. Jerry (RXman) was convinced I needed to rejet and he even found me some leaner #100 and #95 mainjets, but that had zero improved effect: the problem was coming from the lack of atomization of the mainjet. If I rode at 2000 RPM I could ride for 10 miles and the plugs would just be starting to have darkness, but as soon as I revved toward my normal RPM (3k-6k mostly) it would smell of fuel, blow blue smoke, and foul 3 sparkplugs immediately.
When I found this stiffer wire I used it to fight my way thru those passages (5th time the carbs were off, argh...) and all this white powder started showing up: it was by then the only spot I hadn't hand-cleaned (ultrasonics did zero for it). One of them was REAL hard to clear out, it seemed to have really packed itself in, hard. So, it takes patience if you find it in yours.
After that I started doing it with the many carb sets I was getting from others for cleaning: I found it in about 10% of the carb sets, and like mine, not in all of them in that set. Then I did some research to find out 'why' this would happen and came across several reports of 'results of MTBE in fuels' that were done by the EPA and DOT, and some others by people who had left them under protest, and did their own reports. ALL of those documents pointed toward how MTBE had been selected to "...damage zinc-aluminum alloys in the fuel atomization devices..." of the cars' carburetors, intentionally done to get old carb'd cars off the road slowly, but permanently. The last straw (for me) was when the famous reports showed up here and in other States where the government would buy your car ("Cash for Clunkers" program) and then immediately take the ones that ran and would pour "acid mixed with fine sand" into the carbs while running the engine at high speed, until it stopped. Then they were hauled off to crushers. The "acid" was MTBE.