In order for someone to help you with API Request questions, it is critical that you post here the API Request you are using / creating / building.
Remember to replace your Key with [MY_KEY].

What I can anticipate is that it will probably be a two-step operation.
The first step will be to search for and receive the list of movies that fit your criteria / filter.
https://developers.themoviedb.org/3/discover/movie-discover
The second step, now with the IDs, will be to get the details of each movie.
https://developers.themoviedb.org/3/movies/get-movie-details

First of all, you need to know what the ID of each genre is.
https://developers.themoviedb.org/3/genres/get-movie-list
For this you need to have the list that can be received with the following Requisition:
https://api.themoviedb.org/3/genre/movie/list?api_key=[MY_KEY]&language=en-US
For example Comedy = 35

Example for first step:
https://api.themoviedb.org/3/discover/movie?api_key=[MY_KEY]&language=en-US&sort_by=original_title.asc&page=1&primary_release_year=2004&with_genres=35
And you will get something like that:

{
  "page": 1,
  "total_results": 1125,
  "total_pages": 57,
  "results": [
    {
      "vote_count": 204,
      "id": 9805,
      "video": false,
      "vote_average": 6.3,
      "title": "Dreamship Surprise - Period 1",
      "popularity": 7.289,
      "poster_path": "/w4EZC3GeNwLNo5KNkj4QLOWyGII.jpg",
      "original_language": "de",
      "original_title": "(T)Raumschiff Surprise - Periode 1",
      "genre_ids": [
        35,
        878
      ],
      "backdrop_path": "/A41kszh8uig7nWIAgEFdY6MzrLx.jpg",
      "adult": false,
      "overview": "Hundreds of years after humans have settled on Mars, Regulator Rogul and Lord Jens Maul, lead a force of Martians to Earth in order to conquer the planet. Queen Metaphor looks to the gay heroes aboard the spaceship Surprise -- Captain Kork, Mr. Spuck, and first engineer Schrotty",
      "release_date": "2004-07-14"
    },
    {
      "vote_count": 9,
      "id": 59700,
      "video": false,
      "vote_average": 5.4,
      "title": "12 Days of Christmas Eve",
      "popularity": 1.4,
      "poster_path": "/bz9uNLaf9mzopYT6aw6iuCT8E4L.jpg",
      "original_language": "en",
      "original_title": "12 Days of Christmas Eve",
      "genre_ids": [
        10770,
        10751,
        14,
        35,
        18
      ],
      "backdrop_path": "/cUd2qXmRzJEqNcn1XQcGu3CSo3y.jpg",
      "adult": false,
      "overview": "Calvin Carter, a successful business executive, has it all, but neglects those closest to him. On Christmas Eve, all that changes when the sign on his office building falls on him. He awakes in a hospital bed, attended to by Angie, a nurse who soon lets him know he has 12 days (12 chances) to get his act together and achieve the \"perfect\" Christmas Eve. If he doesn't, there will be dire consequences.",
      "release_date": "2004-12-07"
    },
    {
      "vote_count": 2354,
      "id": 10096,
      "video": false,
      "vote_average": 6.5,
      "title": "13 Going on 30",
      "popularity": 9.725,
      "poster_path": "/iSvz2Nk1vzgm7bEMqMazhQ1F3zA.jpg",
      "original_language": "en",
      "original_title": "13 Going on 30",
      "genre_ids": [
        35,
        14,
        10749
      ],
      "backdrop_path": "/1sJoLA3bjKvblPh7GR9URBWZsDO.jpg",

    etc...
    etc...
    etc...

If the details you want did not come in this answer then you will have to make a Request for each ID to receive the details you want.

Thanks for the reply. I was able to work through it by trial and error. The resulting request ended up as follows:

payload = "{}"
URL="/3/discover/movie?with_genres=35&primary_release_year=2004&page=1&language=en-US&api_key=&sort_by=popularity.desc"
conn.request("GET", URL, payload)
res = conn.getresponse()
data = res.read()

I can only help in this part of what you posted:
/3/discover/movie?with_genres=35&primary_release_year=2004&page=1&language=en-US&api_key=&sort_by=popularity.desc
And I think it does not meet the requirements. Compare with the one I posted:

https://api.themoviedb.org/3/discover/movie?api_key=[MY_KEY]&language=en-US&sort_by=original_title.asc&page=1&primary_release_year=2004&with_genres=35  
_________________________/3/discover/movie?with_genres=35&primary_release_year=2004&page=1&language=en-US&api_key=[MY_KEY]&sort_by=popularity.desc

I added the beginning to yours and tested it.
https://api.themoviedb.org/3/discover/movie?with_genres=35&primary_release_year=2004&page=1&language=en-US&api_key=[MY_KEY]&sort_by=popularity.desc
The result is the same as mine. So I think it's settled .