The Great Sphinx of Giza's natural state is buried in sand. Whenever attention towards the Sphinx lapses for long enough the winds of the Giza plateau pile sand deeply around it, leaving mostly just the head. Its first burial probably happened after the end of the Old Kingdom, when the associated temple was destroyed and the Giza complex apparently abandoned. Once in this state it takes a really significant effort to unbury it, which has happened several times. The first of these seems to have been done by Thutmose IV in around 1400BCE as a result of a dream or vision, at which point the Sphinx had been buried and mostly abandoned for some 600 years, though still regarded as a significant religious site. The modern attempt to free the Sphinx started in 1817 but failed several times, only finally completed in the 1930s.

Some work was done on it around 600BCE, in the 26th Dynasty (the Saites), at the beginning of the Late Period, but only on the top and head, suggesting it was at that point buried but still important enough to restore. Herodotus, however, writing around 150 years later, makes no mention of it in spite of his lengthy discussion of the Giza pyramids. There are no obvious signs of major Persian or Ptolemaic restorations. Pliny reports, early in the first century, that the face was then painted red, suggesting that it was still in use as a religious site.

Cleopatra seems to have lived at the tail end of this long period of burial of the Sphinx. The Romans, after taking control of Egypt after Cleopatra's death, made significant restorations and modifications to the Sphinx and its environs after completely freeing it from the sands. It was in fact the Roman temple, walls, and pavements that modern excavators found... and destroyed in order to find older traces.

That the Romans began restorations on the Sphinx after taking over Egypt suggests that the monument was still somewhat important. Presumably in the time of the Ptolemies and earlier, the head was considered of some significance, though not enough to bother with revealing the whole thing. Why the Romans went so far, we don't know; perhaps their efforts in restoring impressive pieces of the Egyptian religion were meant to show how rule by Rome was to be beneficial to Egypt. Or maybe they just thought it would be an awesome tourist attraction. Pliny calls it "still more wonderous" than the Pyramids.

The historian Arrian, in a poetic inscription on the left paw dated 166 CE, suggests a belief that the Sphinx was built after or with the pyramids, and to be guarding a price nearby, which is pretty accurate to what we currently believe, if vague. Pliny claims it to be the burial place of "King Harmaïs" which is not accurate as it was never a tomb.

Cleopatra or any of the Ptolemies could have worked on the Sphinx, or excavated enough to read the Dream Stela, but there's no indication that they did. The sort of suppositions and rumors such as related by Arrian or Pliny are a pretty good place to start as to what Cleopatra might have known (or thought she knew) about the Sphinx. It seems that even at that point all knowledge of its origins was long lost, even though the names of the pyramid builders were preserved.

If she even knew about it at all, that is. The Sphinx seems mostly to have been a minor local shrine, and it's quite possible that the monarch in Alexandria may not have much reason to know or care about it.