Have Homelite Yard Broom II UT08514. Fuel filter fell out of tank when I dumped old gas! Which little black tube (way back in the tank) does it belong on and how do I reconnect it? thanks

look up the unit on the Homelite web site, download the manual, it explains all the steps.

Check the end of the throttle cable at the motor. There should be moment at the carburetor when you advance the throttle.
If you are getting some black smoke check the air cleaner to see if it is plugged with dirt. Has this unit been in winter storage? If so drain and refill the gas tank with fresh fuel.
Regards Outback

Your manual states it is 50:1 which is 2.6 ounces of oil/gallon of gas...or 1.3 ounces in 1/2 gallon of gas. This is a common ratio for Homelite 2 stroke engines. I recommend synthetic oil.
Specs are on page 8 of your manual.
http://dl.owneriq.net/f/f2daf743-4682-4ff5-bb2a-bf9e20e5dfa9.pdf

The manual probably says 40:1 (3.2oz.oil to 1 gal gas) but if you use a good modern 2 cycle oil,50:1(2.6oz .oil to 1 gal gas) will suffice.

.025

It seems that the carb is not getting fuel. It could be a cracked fuel line. The fuel priming bulb could also be cracked/dry rotted. Possibly the fuel pump in the carb is bad. The carb is rather intricate and not something anyone without training should try to repair. The overhaul kit will cost $15 to $20. I suggest you take the blower to a repair shop.

could be fuel lines, fuel filter, or your carburetor needs to be rebuilt. The carb has thin rubber diaphragms that get hard over time, hot weather, or stale gas. Need more assistance,
email [email protected].

HM 270021002 throttle cable $2.60 + 10.50 shipping. Can have it to you in 2 business days. Would you like an invoice? if so, reply to [email protected]

I’m happy to assist further over the phone at https://www.6ya.com/expert/rolandw_c013176d62ee17ae

You will need part# 270021002.
Good Day.

"top of the gas tank where the gas tube fits into the tank"

I assume you're describing the gas tank cap. if so, the o-ring is missing from the gas tank cap. You can find those at hardware stores.