Like the famous "Om Jayanti Mangla Kali Bhadra Kali Kapalini" verses.
ॐ जयंती मंगला काली भद्रकाली कपालिनी
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The quoted verse is from the Argala stotra, a part of the Devi Mahatyam of the Markandeya Purana.
जयन्ती मङ्गला काली भद्रकाली कपालिनी ।
दुर्गा शिवा क्षमा धात्री स्वाहा स्वधा नमोऽस्तु ते ॥२॥
Jayantii Manggalaa Kaalii Bhadrakaalii Kapaalinii |
Durgaa Shivaa Kssamaa Dhaatrii Svaahaa Svadhaa Namostu Te ||2||
2.1: Salutations to Jayanti (Who is Ever-Victorious), Mangala (Who is the bestower of Auspiciousness), Kali (Who is beyond Kala or Time), Bhadrakali (Who is the controller of Life and Death, being beyond Kala or Time), Kapalini (Who wear a Garland of Skulls), 2.2: Salutations to Durga (Who is Durgati-Nashini), Shiva (Who is Ever-Auspicious and One with Shiva as His Consort), Kshama (Who is an embodiment of Forbearance), Dhatri (Who is the Supporter of all Beings), Swaha (Who is the final receiver of the Sacrificial Oblations to Gods) and Swadha (Who is the final receiver of the Sacrificial Oblations to Manes); Salutations to You.
Now, according to the following verse from Vyasa Smriti, Upanita twice-borns (i.e. Dvijas who have received Upanayana Samskara) are already qualified to read Puranas.
Brahmanas, Kshatriyas, and Vaishyas are called the twice-born. Their first births take place when they are delivered of their mother's womb ; their second, when they duly accept the Gayatri Mantra from their preceptors (21).
Thus made twice-born, and free from all other faults, they become entitled to study the Vedas, Smritis and Puranas (22).
Therefore it appears that an Upanita Dvija can recite the given Shloka/Mantra without any further initiation. For others I'm not sure.
(NOTE: It's only a partial answer)