For all the mahAvidyA goddesses, there is an associated Bhairava or Shiva, whose upAsanA is considered mandatory by some sources. The Bhairavas for the ten mahAvidyA-s are:
kAlI - mahAkAla bhairava
tripurasundarI – laliteshvara bhairava (or tripureshvara bhairava)
tArA - akShobhya bhairava
ChinnamastA - karAla bhairava (or krodha bhairava)
bhuvaneshvarI - mahAdeva bhairava
dhUmAvatI - kAla bhairava (or ghora bhairava)
Mahalakshmi - nArAyaNa bhairava (or sadAshiva bhairava)
tripura bhairavI – vaTuka bhairava
mAtangI - matanga bhairava
bagalAmukhI - mrityunjaya bhairava
strImantro bhogadaH proktaH pummantro mokShadaH param |
ubhayopAsanaM devi bhuktimuktipradAyakam ||
upAsanA of mahAvidyA grants bhoga and when done along with the corresponding Bhairava/Shiva, grants Bhoga and Moksha.
kevalaM yo japecChAktaM manuM shaivaM na yojayet |
janmakoTishu japto.api na manuH siddhibhAg bhavet ||
Without the upAsana of the respective Bhairava/Shiva, reciting the female mahAvidyA alone does not grant one Mantra Siddhi. However, in the case of Srividya, the mantra is both a mantra and a vidya as it is shiva-shaktyAtmaka:
Ka-trayaM ha-dwayaM chaiva shaivo bhAgaH prakIrtitaH etc.
Thus, among the Dasha Mahavidyas, the need for reciting a separate mantra for Tripureshwara Bhairava is eliminated for Sundari as Srividya itself denotes both Shiva and Shakti. Umanandanatha prescribes kAmeshwara mantra as an anga for mahAShoDashi but this is not in line with Kalpasutras, which has no mention of this mantra. This mantra seems to be present only in trikUTA rahasya, an appendix to Rudra Yamala, which is not a standard Sri-Kula Tantra. Moreover, the Rajarajeshwari Parishishtha and Shodashi Prakarana of the same Yamala has no mention of Kameshwara Mantra. Also owing to the fact that trikUTA Rahasya is solely addressing vAma mArgi-s (kApAlika-s and kShapaNaka-s in particular), others can safely neglect this mantra and Umanandanatha's prescription, which is not in accordance with the Kalpasutra. Again, the same rule applies for hAdi vidyA as well. Though Devata for hAdi vidyA is mahA kAmeshvara bhaTTAraka, the mantra, like kAdi, is again shiva-shaktyAtmaka (pramANa is from hAdi mata and vAmakeshwara tantra) and thus, the need for a separate kAmeshvarI mantra is eliminated. As Srividya is shiva-shaktyAtmaka, it grants both Bhoga Moksha by upAsana of one single mantra. Hence, bhoga moksha pradAtrtva of Srividya is highlighted among the ten mahAvidyAs. However, even other mahAvidyAs can be considered shiva-shaktyAtmaka by the presence of mAyA bIjA - which is ubhayAtmaka. This however does not apply to dhUmAvatI, where the mAyA bIja is missing, indicating mithuna-hInatva. Hence, some Sampradayas do not insist on the mandatory upAsana of the corresponding Bhairava.
Guru/svashAstra is the final authority here.