This is not quite correct.
First, you can change the exponent in the law of gravitation from -2 to -(2 +
a), where a is a small number. Secondly, there's the possibility of previously
unobserved masses which might cause the perihelion precession, in the way that Neptune
caused the previously unexplained departures of Uranus from its theoretical orbit.
Leverrier, one of those who predicted Neptune, suggested that there might be a planet
"Vulcan" inside the orbit of Mercury to cause the latter's perihelion precession. The
possibility that there might be a continuous distribution of dust (such as that which
causes the Zodiacal light) of sufficient density to have the necessary gravitational
effect, was also considered.
There's always wiggle room. The real question is whether or not the wiggles
that are needed to save a theory have any plausibility. That's partly a matter of
taste. Changing the exponent in Newton's law with no other justification is just ugly.
But - especially with the succesful prediction of Neptune - the Vulcan or dust
hypotheses had to be, & were, taken very seriously. Of course their plausibility
declined as people made observational searches for these things & didn't find them.
George L. Murphy
gmurphy@raex.com
http://web.raex.com/~gmurphy/