Cracked it!
As expected, I was entering the Power Query syntax wrong for the Web.Contents() function. The correct syntax should be:
let
Source = Json.Document(Web.Contents("https://kc.humanitarianresponse.info/api/v1/data/814220",[Query=[query="{""$and"": [{""assessor_details/date_of_assessment"": {""$gte"": ""2021-09-01""}},{""assessor_details/date_of_assessment"": {""$lte"": ""2021-09-03""}}] }"],Headers=[Authorization="token XXXXX"]])),
#"Converted to Table" = Table.FromList(Source, Splitter.SplitByNothing(), null, null, ExtraValues.Error)
in
#"Converted to Table"