The line of the song quoted is too fragmentary to identify the song with any specificity, at least not without some fairly intensive research. But it's evidently a song about Krishna.
In the Bengali original of My Boyhood Days, ছেলেবেলা, Tagore gives the first line of this song as:
ময় ছোড়োঁ ব্রজকী বাসরী / may chho.Do.N bR^ij_kii baasarii
That is not Bengali. It is likely the dialect of Hindi called Braj Bhasha, spoken in the Braj region where Krishna spent his childhood, roughly transliterated into the Bengali script. A grammatical Braj Bhasha sentence that could be the original of that transliteration could be:
मैं छोड्यो बृजकी बाँसुरी / mai.n chhoDyo bR^ij_kii baa.Nsurii
This means "I have left the flute of Braj". The power of Krishna's flute to enchant listeners is a common theme in the Indian tradition. It is said to enrapture not just all human listeners, but also animals and birds. Without more of the lyrics, it's hard to narrow down exactly which song this could be, because there are literally thousands of songs about Krishna and his flute.
More specifically, there are many songs that take as their theme the lover's quarrels, or sometimes just the teasing, between Krishna and his consort Radha around the flute. In the 16th century, both Meera and Surdas wrote poems on this theme, for example:
राधा प्यारी, दे डारो ना बंसी मोरी / raadha pyaarii, de Daaro naa ba.nsii mori
"Radha my love, do give me back my flute" (Meera);
कान्हा तोरी बंसरी नेक बजाऊँ / kaanhaa torii ba.nsarii nek bajaauu.N
"Krishna, let me play your flute" (Surdas).
Both these poems have been set to music any number of times through the centuries.
So "mai.n chhoDyo bR^ij_kii baa.Nsurii" could be the beginning of a song where Krishna placates Radha, asking her not to be jealous of or angry about his flute because it's at his lips, which means he's kissing the flute and not her; or because it attracts other women to him; or because his entrancing playing keeps her from her work. He's mollifying her by saying, "Look, I've left off playing the flute".
It could also be a lament. At some point Krishna had to leave Braj and childish things and move to Dwarka. The adult, royal Krishna isn't particularly associated with the flute, as it's what he used to do to while away the time when he was a cowherd. So the opening line could be that of a song where Krishna is going to Dwarka as an exile from Braj. However, that would be rather an unusual theme for a song, and the likelihood is vanishingly small when compared to that of its being a song on the Krishna-Radha theme.
The love of Krishna and Radha was the chief subject of the Vaishnava Padavali poetic tradition of that flourished around the same time as Meera and Surdas. Tagore was very drawn to this tradition as a young teenager. One of his first poetic works, Bhanusingher Padavali, was a pastiche in the Brajabuli dialect that he passed off as an authentic text from that tradition. So another possibility is that the song Tagore recalls Srikantha singing is a Vaishnav Padavali poem in Brajbuli rather than Braj Bhasha (the two are quite distinct). Without more of the lyrics, one can't tell.