Here’s how you could do that (inspired by this). Make sure to check the Content-Type of the response (as shown below), so that you can modify it by adding the metadata, only if it is of application/json type.

For the OpenAPI (Swagger UI) to render (both /docs and /redoc), make sure to check whether openapi key is not present in the response, so that you can proceed modifying the response only in that case. If you happen to have a key with such a name in your response data, then you could have additional checks using further keys that are present in the response for the OpenAPI, e.g., info, version, paths, and, if needed, you can check against their values too.

from fastapi import FastAPI, Request, Response
import json

app = FastAPI()

@app.middleware("http")
async def add_metadata_to_response_payload(request: Request, call_next):
    response = await call_next(request)
    content_type = response.headers.get('Content-Type')
    if content_type == "application/json":
        response_body = [section async for section in response.body_iterator]
        resp_str = response_body[0].decode()  # converts "response_body" bytes into string
        resp_dict = json.loads(resp_str)  # converts resp_str into dict 
        #print(resp_dict)
        if "openapi" not in resp_dict:
            data = {}
            data["data"] = resp_dict  # adds the "resp_dict" to the "data" dictionary
            data["metadata"] = {
                "some_data_key_1": "some_data_value_1",
                "some_data_key_2": "some_data_value_2",
                "some_data_key_3": "some_data_value_3"}
            resp_str = json.dumps(data, indent=2)  # converts dict into JSON string
        
        return Response(content=resp_str, status_code=response.status_code, media_type=response.media_type)
        
    return response


@app.get("/")
def foo(request: Request):
    return {"hello": "world!"}

Update 1

Alternatively, a likely better approach would be to check for the request’s url path at the start of the middleware function (against a pre-defined list of paths/routes that you would like to add metadata to their responses), and proceed accordingly. Example is given below.

from fastapi import FastAPI, Request, Response, Query
from pydantic import constr
from fastapi.responses import JSONResponse
import re
import uvicorn
import json

app = FastAPI()
routes_with_middleware = ["/"]
rx = re.compile(r'^(/items/d+|/courses/[a-zA-Z0-9]+)$')  # support routes with path parameters
my_constr = constr(regex="^[a-zA-Z0-9]+$")

@app.middleware("http")
async def add_metadata_to_response_payload(request: Request, call_next):
    response = await call_next(request)
    if request.url.path not in routes_with_middleware and not rx.match(request.url.path):
        return response
    else:
        content_type = response.headers.get('Content-Type')
        if content_type == "application/json":
            response_body = [section async for section in response.body_iterator]
            resp_str = response_body[0].decode()  # converts "response_body" bytes into string
            resp_dict = json.loads(resp_str)  # converts resp_str into dict 
            data = {}
            data["data"] = resp_dict  # adds "resp_dict" to the "data" dictionary
            data["metadata"] = {
                "some_data_key_1": "some_data_value_1",
                "some_data_key_2": "some_data_value_2",
                "some_data_key_3": "some_data_value_3"}
            resp_str = json.dumps(data, indent=2)  # converts dict into JSON string
            return Response(content=resp_str, status_code=response.status_code, media_type="application/json")

    return response

@app.get("/")
def root():
    return {"hello": "world!"}

@app.get("/items/{id}")
def get_item(id: int):
    return {"Item": id}

@app.get("/courses/{code}")
def get_course(code: my_constr):
    return {"course_code": code, "course_title": "Deep Learning"}

Update 2

Another solution would be to use a custom APIRoute class, as demonstrated here and here, which would allow you to apply the changes on the response body only to routes that you specify—which would solve the issue with Swaager UI in a more easy way.

You could still use the middleware option if you wish, but instead of adding the middleware to the main app, you could add it to a sub application—as shown in this answer and this answer—that includes again only the routes for which you need to modify the response in order to add some additional data in the body.