This paper is probably relevant for your question:
MR0528966 
Wolpert, Scott
The length spectra as moduli for compact Riemann surfaces.
Ann. of Math. (2) 109 (1979), no. 2, 323–351.

And here is a more recent paper on the subject:

MR3770180 
Parlier, Hugo
Interrogating surface length spectra and quantifying isospectrality. (English summary)
Math. Ann. 370 (2018), no. 3-4, 1759–1787.

These papers suggest that finitely many geodesic lengths never determine the surface.