How do we not know what's in the Solar System? It seems every once in a while there's a claim of a new planet in the Solar System.

I don't understand how local planets can stay undiscovered for this long when astronomers are accomplishing what seem to be far more difficult feats.

For example, it's been a whole century since they had accurate enough measurements to tell that Mercury was precessing in a way that didn't jive with Newtonian mechanics.
Now, if there are Earth-sized planets out there, shouldn't their presence have a pretty sizable effect on the orbits of bodies such as Neptune and Pluto? How can there be such large planets with no measurable effects until 2017 technology came along?
There could be another planet in the solar system that has not been discovered if that planet had a highly elliptical orbit with a single orbit taking many hundreds of earth years to complete. That planet could be in the part of its orbit further from sun and could stay in that area for a few hundred years, longer than we've had large telescopes.

At such a distance it would receive very little sunlight and the chances of visually detecting it would be very small. And we wouldn't know where to look either, which just compounds the problem.

Other more distant plants have been discovered but their discovery was due to the effects on the star or stars it is orbiting. We observe a planet passing in front of a star which briefly lowers the apparent magnitude of its parent star. This is a good clue of a planet in orbit. This is impossible in our solar system because an unknown planet would have to come between us and the sun. We would easily see it if that were the case.

We have observed some discrepancies in the orbits of some of the outer planets. This has led to the speculation of another planet. It is very difficult to determine the cause of these small discrepancies though. All bodies orbiting our sun are not only effected by the suns gravity but by the gravitational attraction of all other bodies (including all objects in the asteroid and Kuiper belts). This is a very intense calculation. This attraction constantly changes due to the motion of all the bodies.

Simulations have been developed to study this but must take a lot of computing time even with today's high speed computers.

I hope this helps explain the situation.