Is there such a thing as an ephemerides equation?

The plural form of the word "ephemerides" is "ephemeris," which provides the positions of astronomical bodies as tables that could be used to look up positions.

These days, modern ephemerides are sophisticated computer representations of positions, with the data consisting of coefficients of polynomial equations that provide the coordinates of a body at a given time. The coordinates are computed by evaluating the polynomials for the time the position is required, with time being measured in Julian centuries from the epoch. The time origin is typically J2000 - 2000-01-01 12:00:00.

An ephemeris, then, is three or more complex time polynomials for each body; it is not an equation per se.

Yes, the ephemeris equation corrects orbital parameters to enhance the precision of predicting celestial body positions and is used in astronomy to account for differences between predicted and observed positions of celestial objects.