Did Sai Baba's Guru ever give him a secret mantra? And why doesn't Baba give out mantras himself?

No, Sai Baba's Guru never gave him a secret 'ear-mantra'. As recounted in Chapter 19, Baba had a strong desire to receive a mantra and served his Guru for twelve years, yet his Guru never gave him one. Instead, the Guru's teaching and protection were conveyed through his loving gaze, much like a mother tortoise nurtures her young from across a river just by sight. Because of this, Baba explains to a devotee that he himself cannot give a mantra or 'upadesh' (instruction) in that way. He states, "my Guru did not blow any mantra into my ears, so how can I blow one into yours?" His method, instead, is to have the devotee simply focus their attention on him for spiritual wealth to manifest.