From the description, it sounds like the neutral may be shared between the two circuits, which you can't do with a GFCI. The GFCI is looking for an imbalance of current on the hot and neutral lines, so if the neutral is shared with another circuit, there will be some current returning on the neutral that did not go out on the hot.
If you can trace the circuit and find the connection between the neutrals on the two circuits, and isolate them, that would be the easiest and cheapest fix. If the neutral is shared via a multiwire branch circuit from the breaker and can't be split, you can change the gfci breaker to a standard breaker and install a gfci outlet after the neutral splits for the two branches. And I'm also seeing two pole gfci breakers which would allow you to replace the two breakers with one that provides gfci coverage to both circuits and should handle the shared neutral, which may be a good fall back option if you can't locate the shared neutral.
You'll need a double-pole GFCI breaker, wired like so:
![enter image description here](https://i.stack.imgur.com/OlXRG.png)
What you have described is a multiwire branch circuit. Because the neutral is shared, the overcurrent protection device(s) (in your case the two separate breakers) must be able to trip together.
This can be remedied by installing a double-pole breaker, or by handle-tying two individual breakers. However, since you already have a need for GFCI protection, go ahead and install a double pole GFCI breaker.
However, existing lights might not need GFCI protection if they are 5 ft or more away from the pool (horizontally) and 5 ft or more above the maximum water level of the pool. Within 5 ft and below 5 ft above the maximum water level is not allowed. See (B)(3) and (4).
If you move or install new lights, they are subject to new installation requirements. See (B)(1).
2014 NEC
680.22 Lighting, Receptacles, and Equipment.
(B) Luminaires, Lighting Outlets, and Ceiling-Suspended
(Paddle) Fans.
(1) New Outdoor Installation Clearances. In outdoor pool areas,
luminaires, lighting outlets, and ceiling-suspended (paddle) fans
installed above the pool or the area extending
1.5 m (5 ft) horizontally from the inside walls of the pool shall be installed at a height not less than 3.7 m (12 ft) above the maximum
water level of the pool.
(3) Existing Installations. Existing luminaires and lighting outlets
located less than 1.5 m (5 ft) measured horizontally from the inside
walls of a pool shall be not less than 1.5 m (5 ft) above the surface
of the maximum water level, shall be rigidly attached to the existing
structure, and shall be protected by a ground-fault circuit
interrupter.
(4) GFCI Protection in Adjacent Areas. Luminaries, lighting outlets, and ceiling -suspended (paddle) fans installed in the area extending between 1.5 m (5 ft) and 3.0 m (10 ft) horizontally
from the inside walls of a pool shall be protected by a ground-fault
circuit interrupter unless installed not less than 1.5 m (5 ft) above
the maximum water level and rigidly attached to the structure adjacent
to or enclosing the pool.
Best Answer
Perfect Solution
The GFCI in the breaker is all you need, so replace the other GFCI receptacles (whether inside or out) with regular receptacles and you will be fine. Except, as noted by Harper, the receptacles that need GFCI protection (and have it but "elsewhere") need to be marked GFCI protected.
There are three reasons why there may be GFCI receptacles in addition to the GFCI breaker:
Previous Owner Confusion - Previous owner didn't understand how things work and had GFCI deliberately installed in both breaker panel and receptacles for perceived extra safety.
Home Inspector or Real Estate Agent - A well-meaning, but not knowledgeable, home inspector or real estate agent told the previous owner to put in the outside GFCI in order to meet current code, without understanding that the GFCI in the breaker took care of everything safely
Retrofit - GFCI receptacles originally installed where needed to meet current code. At a later time, the breakers were upgraded (either an entire panel replacement or individual breakers replaced with GFCI breakers) to include GFCI, but the GFCI receptacles were never removed.
In any case, you can get rid of the GFCI at the receptacles. They are more vulnerable to weather-related problems (even with weather-resistant covers) than regular receptacles. If moisture really does get into the receptacles in a way that causes a ground fault, the GFCI breaker will protect you.
The only advantage of a GFCI receptacle over a breaker is that it is convenient to reset. However, there is no guarantee as to whether which GFCI (receptacle or breaker) will trip first - or they may both trip at the same time - so that advantage goes away when you have GFCI in more than one location.