Ramp secret sharing replaces the random coefficients in original Shamir secret sharing with original data to be shared. In such a case, all the coefficients in the polynomial evaluation will be secrets. If such ramp secret sharing supports additive homomorphism, how these secrets are reconstructed. Can the coefficients other than constant term can be reconstructed through Lagranage interpolation.
It does indeed. I've written a bit about this here with code examples: https://mortendahl.github.io/2017/06/04/secret-sharing-part1/#packed-variant.