The flyby anomaly is a discrepancy between current scientific models and the actual increase in speed (i.e. increase in kinetic energy) observed during a planetary flyby (usually of Earth) by a spacecraft. In multiple cases, spacecraft have been observed to gain greater speed than scientists had predicted, but thus far no convincing explanation has been found. Source Wikipedia
The speed increase is relatively small, at about 3 to 13 mm/s, nevertheless, Investigations by the US Jet Propulsion Laboratory, The Goddard Space Center and the University of Texas have not been able to explain the anomalies, which have now been observed and verified several times, with at least 10 different spacecraft. (first observation 1990).
Possible theories include : the presence of Dark Matter, lightspeed variations and gravity variations.