I have the following sheet music excerpt for Spanish guitar:
What is the meaning of the ₵2 and C3 on the 2nd and 4th measures?
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The symbols in question represent in fact barre chords. The numerals following the C indicates the position (i.e., fret) of the barre chord. As OP already has figured out, ₵ refers to a partial barre chord—it does, however, not indicate how many strings should be barred.
A nice overview of the many different styles to notate barre chords is given in the respective Wikipedia article.