Is there a way to play the note that is halfway between C and C#?

[Perry]

It's a great sound that old-timey between the cracks playing. In the blues it is not a perfect flatted third either. Compton can approximate that third with those "Monroe like" down slides.

[mandocrucian]

Just BEND the note by pushing the strings toward the bass strings (if you are on the A strings.)

If it's the low "neutral C" (on the G strings) you want, you'll have to pull the strings toward the treble strings, which is a little harder. (If you push, you'll probably push the strings off the fretboard)

It's not even a full 1 fret (half-step) bend! If you can't manage that, start doing some fingertip pushups and increase your finger strength.

[Tom Sanderson]

You should listen to David Grisman bending his mandolin strings sometime. The tension on mandolin strings may high, but he has real finger power. So YES, you can bend mandolin strings! Get to C# from C? No problem!

Tim O'Brien too.

[foldedpath]

Just BEND the note by pushing the strings toward the bass strings (if you are on the A strings.)

If it's the low "neutral C" (on the G strings) you want, you'll have to pull the strings toward the treble strings, which is a little harder. (If you push, you'll probably push the strings off the fretboard)

It's not even a full 1 fret (half-step) bend! If you can't manage that, start doing some fingertip pushups and increase your finger strength.

Just repeating something that I think was mentioned earlier. Bending to a "C Supernatural" works in some genres of music, where you have the time to get up there and back down. It doesn't work in a fiddle tune at dance tempo, where that single note is in the middle of a flurry of other notes at full speed.

In that case, I'll still play the note -- no other options! But I'll avoid voicing chords that conflict with it.