Príomh- is a prefix that means prime, principal, main ⁊rl…
Oifig is a feminine noun that means office.
Thus, príomhoifig: main office, head office.
The addition of a prefix does not affect the gender of the noun. Therefore, since oifig is feminine, the initial p of príomhoifig would be lenited when preceded by the definite article: an phríomhoifig (the main office).