Montu
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Sounds like JohnT has the simplest solution. Just use powerline networking. I have not had a need to do this yet but I believe the technology has advanced and will work. When others are talking about POE are you talking about Power Over Ethernet? If you are doing power over ethernet then you are already running an ethernet cable so problem solved. I apologize in advance if this is not the POE mentioned.
PowerLine adapters should be avoided when possible, the tech has gotten better and the performance is now acceptable but the security on these range from poor to weak.
Link on poor security: Vulnerability: Infiltrating a network via Powerline (HomePlugAV) adapters - www.bentasker.co.uk
PoE is Power over Ethernet, the line of thinking is you run this the closest you can get and put an access point there (if you can't make it to the pergola the closest outer wall of the house would probably work), if you have devices that can be wired at that location then sure add in some more runs and wire those..but I assume you'll want to use wifi devices out there so you'll need the AP.