The physics of turn coordinators, and turn rate indicators, are complex.
The since the gyro axis is not usually not perfectly horizontal, the instrument shows some response to pitch rate, as well as to turn rate. In a coordinated turn at some particular turn rate, the calibrated markings can only be perfectly accurate at one specific pitch rate. This means we have to specify either the bank angle or the True airspeed when we calibrate the instrument for a given turn rate. In practice, the True airspeed is specified and assumed to be constant regardless of turn rate, while the bank angle, naturally, varies with turn rate.
Errors from flying at a different True airspeed than the instrument was calibrated for may be small enough to generally ignore in actual practice, but they do exist.
This is explored in more detail in the following ASE answers to related questions--
May turn rate indicators (and turn coordinators) be calibrated to be accurate at all airspeeds, or only for one?
Are turn rate indicators calibrated to a certain speed? If so, what speed?
In some "turn coordinators", does the gyro spin opposite to the direction it spins in a "turn rate indicator"?
If the aircraft is slipping or skidding, then the relationship between True airspeed, bank angle, pitch rate, and turn rate is altered, so the calibrated markings will no longer be accurate even if the True airspeed is the one that the instrument is calibrated for. For example, in a wings-level skidding turn, the pitch rate is zero. For the reasons explored in the above links, this will create some change in the turn rate indication. The indicated turn rate will be different than if we were flying at the same True airspeed but in a fully coordinated turn.
Another way to look at it is that all turn rate indicators, and some if not at all turn coordinators, are designed so that the gimbal axis of the gyro stays approximately horizontal relative to the earth during any given coordinated turn. If the aircraft is not banked at all, then any turn will tilt the gimbal axis of the gyro away from horizontal, which will affect the sensed turn rate.
Of course, it could happen that a change in indication due to slipping or skidding could exactly offset a change in indication due to flying at some True airspeed that is different from the True airspeed that our instrument is calibrated for. So the full answer to the question is "Generally, yes, but it's complicated."
The details discussed in the third link above suggest that there may be fundamental differences in the way that a skid, or a slip, affects the reading of a turn rate indicator versus a turn coordinator--i.e. whether the indicated turn rate is increased or decreased-- and furthermore that there may be differences in behavior between different makes of turn coordinator, because they may not all share the same direction of spin of the gyro wheel.
This would be fertile ground for some actual in-flight experiments to make some real-world observations as to how the turn rate displayed by a given instrument is altered by slipping or skidding. Keep the airspeed and the indicated turn rate constant, and make the aircraft slip or skid through one or more full 360-degree turns, and measure the actual time to complete 360 degrees of turn in each case. For completeness, repeat the experiment at a different airspeed, or a different indicated turn rate, or a different amount of deflection of the slip-skid ball. Be sure to include the extreme case of a wings-level skidding turn. Finally-- especially if flying with a turn coordinator rather than with a turn rate indicator-- do some basic tests to see if, starting from a standard-rate turn, moving the stick or yoke aft to "pull" some extra G's tends to boost, or to reduce, the indicated turn rate. Similarly, starting from a standard-rate turn, test whether moving the stick or yoke forward to "unload" the wing to a lower G-load tends to boost, or to reduce, the indicated turn rate. These tests will help show which direction the gyro wheel is spinning in that particular instrument, which is relevant to understanding what is going on in the slipping or skidding situations.
