You can feed 5V directly into the 5V pin of the Arduino as ChrisL says. However, that bypasses the protection circuits built into the Arduino. If you ever forget to disconnect the 5V and plug a USB cable into your Arduino then you risk damaging it, since both your external power supply and the USB port will be feeding 5V into the 5V line.
What I decided to ...
I assume, that your problem is how to connect everything to the power supply.
You have some different options, depending on your requirements. The linked adaptor (barrel plug to alligator clips) doesn't seem to fit the barrel plug of the power supply. So you cannot use that. Especially it seems, that both are the male part of the connection. For every plug ...
If you can upload sketches, I don't think it is that big of a deal but if you can't, check the TX and RX pin tracks since if those tracks are shorted, they can keep the indicator lights on for ever due to the Serial code on the Arduino. To get further help, please tell me what happens if you try uploading code.
@ Duncan C
think i have a solution not sure if it's your way but it does work so not bad for a newbie lol
// Gentle, blue-green ocean waves.
// December 2019, Mark Kriegsman and Mary Corey March.
// For Dan.
// Thanks for the sketch i have added a daylight sensor so they now go off in daylight Dave Cowell June 2020
An LED wants positive voltage to flow from the anode to the cathode (the + terminal to the - terminal)
You can wire it to a digital pin 2 different ways. (In both cases you'll also need a current limiting resistor to avoid burning out your LED and/or your digital pin, but ignore that for this discussion.)
You can wire the positive terminal of the LED to +5V, ...
You're unlikely to find exactly what you want as existing code example. Break it into parts and solve it in parts.
You can't "Run a sketch" based on reading a value of an LDR. You can write a new sketch that reads an LDR, and calls a function if it's dark, or a different function if it is light. You will need to refactor your code from your other ...
I'm not sure exactly what you want, but if you expect after a single command on to have it looped 100 times, you need to call the redToBlue function in every loop (within wait_command, that checks for a boolean if the on command has been requested, and when so, call the redToBlue function. When the off command is set, you reset the boolean.