confused about power

You need to be a bit more specific about your hardware building blocks to answer your questions. For example: Pico drives a driver that drives the relays... but what is in between the Pico and the relays boards with 4 relays each?

Decoupling capacitors are a good idea... but often it is not good enough... decoupling or actually have 'separate' power supplies for the different (types of building blocks) is a good idea. For example, want to have a very stable and clean power supply for your control electronics, such as the pico - and the chip sitting on the breadboard in the power strip. I assume it is a shift register or port expander to drive the 8 relays (it cannot be a ULD2003 driver, because this driver has only 7 outputs... leads me to the question: how many relays do you control? Last but least your cat cable from Pico to this 'mysterious' chip has only 8 leads, which allows you to control directly - or parallel - only 7 things, because one of these 8 MUST be the shared ground. This is a good segue way to talk about shared Ground: shared ground is the only way if you do not use opto coupling to have galvanic/electrically separate circuits. Shared Ground is a good starting point, no so good is shared power or insufficiently decoupled power supply. Next would be to look how you get the power from the black box to the strip for the 'mysterious' chip and the relays boards. If it is the cat as well, then you can drive 6 relays only... on the other hand, you may have extra leads for getting the power to the 'mysterious' chip and the relays boards.

On a different note: Each relays consumes about 30mA... and that can bother your 'power world' too...

Using batteries is a good test stepping stone to figuring out the problems... but the batteries do not last long... except you make them rechargeable and somehow recharge them. In that case, the act like large decoupling capacitors and smoothen out the power as well as buffer it for the peaks when you need more than the power supply can stably provide.

First you start of with one relays and strip unpowered to test functionality. Assuming this works, you add the rest of relays and do some testing. With a Multimeter you can measure the relays contacts (if the relays boards do not have LED(s) on the board to show on or off status.

For the power supply, you have to be sure that your power supply pushes enough power for all your devices... I see some additional 'things' in your 'black box'. If your power supply delivers 5 volts and is strong enough to run all things, you can create a decoupling: use a diode (Anode, not-ring side) from plus to (Cathode, ring-side) to PICO Bat and and a large capacitor of 1000uF from diode Cathode/Pico Bat to Ground, that stabilizes PICO board input power pretty well. You do even more and add a capacitor between Pico 3.3V and Ground. You can also have the latter capacitor... but it is a bit a stress on power up for the regulator if the capacitor is big. Both is for sure better.
What the diode does is preventing back flow of power from circuit power buffered with capacitor from pico (if you put only the second capacitor - 3.3V to ground), the regulator works about the same way as the diode... but the 'farther away' from the actual MC you have stabilization, the better. Until you say what your 'mysterious' chip is, I cannot tell anything about how to power it... 3.3V from pico? 5V direct form power supply?...

The relays board can usually have their own supply because they haven an open-collector kind of driver on them, which means, as soon as you put TTL-High minimum - 2 Volts - on the input, the switch this alternate power to the relays coil and do the on switch, the a bit more than 2V is better, but not more than power supply voltage of the driver). Btw, TTL-LOW is max 0.8 Volts. The minimum of 1.2 Volts difference is to prevent noise from switching... Some drivers need less to switch on but are then also more susceptive to noise. To all that sharded ground applies.

Take my comments with pinches of salt... I spent the majority of my time in software as well... btw, would be interested to see some of your control software published here...