Today my Startup Manager gave me the following message:
“The program 20131224 has registered the executable C:\Program Files\Avast Software\Avast\setup\emupdate\f78bd4a0-0494-4061-8be6-4767dc3031ec.exe/check to run at system startup. Do you wish to allow this change?”
Is this legitimate? If it’s legitimate and I already told it “No” - will it ask again or what should I do to allow it?
Its an Emergency Update and legit, so allow it. 
Yes, it is legitimate, it is a file created by the avast Emergency Update function.
I honestly don’t know if it would ask again, as what it is asking about what is the creation of a RunOnce registry entry, rather than allowing the file to run as such.
When avast starts there is a file AvastEmUpdate.exe which does a check to see if there is an emergency update available. If there is then it creates a uniquely named file and creates the RunOnce entry so that it runs the next time you boot to install the update.
So I would say that it is likely that the AvastEmUpdate.exe file would try to recreate the RunOnce entry. That said I guess you could double click on the file it created and that may be enough to get the emergency update installed.