oltv00

Как то так.
Не уверен что вызвал тот метод dataTaskWithRequest
Но он тоже должен сработать в этом случае.

#import <AFNetworking.h>

    NSString *urlString = @"https://api...";
    NSURL *url = [NSURL URLWithString:urlString];
    NSMutableURLRequest *mutableURLRequest = [NSMutableURLRequest requestWithURL:url];
    NSMutableData *requestBodyData = [NSMutableData data];

    mutableURLRequest.HTTPBody = requestBodyData;
    mutableURLRequest.HTTPMethod = @"POST";

    NSString *accessToken = @"accessToken";
    NSString *value = [NSString stringWithFormat:@"Bearer %@", accessToken];
    [mutableURLRequest setValue:value forHTTPHeaderField:@"Authorization"];

    [mutableURLRequest setValue:@"application/json" forHTTPHeaderField:@"Content-Type"];


    [[AFHTTPSessionManager manager] dataTaskWithRequest:mutableURLRequest completionHandler:^(NSURLResponse * _Nonnull response,
                                                                                              id  _Nullable responseObject,
                                                                                              NSError * _Nullable error) {
        //We do not reach this code block !

        // Save the incoming data to CoreData
        completionHandler(UIBackgroundFetchResultNewData);
    }];