mocha already supports asynchronism. Here is the reference code

var myModule = require('../') //
describe('myModule', function () {
    it('onSuccess', function (done) {
        myModule(function(){done()},function(){done('')});
    }
    it('onError', function (done) {
        myModule(function(){done('')},function(){done()});
    }
}

main.js :

module.exports = function(onSuccess, onError) {
  var p = new Promise(function(resolve, reject) {
    setTimeout(function() {
      resolve('ok');
    }, 1000);
  });

  p.then(function(response) {
    onSuccess(response);
  }).catch(function(error) {
    onError(error);
  });
};

main.test.js :

const request = require('./main');
const sinon = require('sinon');

const flushPromises = () => new Promise(setImmediate);

describe('1010000015955091', () => {
  let clock;
  before(() => {
    clock = sinon.useFakeTimers({ toFake: ['setTimeout'] });
  });
  after(() => {
    clock.restore();
  });
  it('should success', async () => {
    const onSuccessStub = sinon.stub();
    const onErrorStub = sinon.stub();
    request(onSuccessStub, onErrorStub);
    clock.tick(1000);
    await flushPromises();
    sinon.assert.calledWithExactly(onSuccessStub, 'ok');
  });
});

Unit Test results + Test coverage report:

  1010000015955091
     should success


  1 passing (11ms)

----------|---------|----------|---------|---------|-------------------
File      | % Stmts | % Branch | % Funcs | % Lines | Uncovered Line -sharps 
----------|---------|----------|---------|---------|-------------------
All files |   85.71 |      100 |      80 |   85.71 |                   
 main.js  |   85.71 |      100 |      80 |   85.71 | 11                
----------|---------|----------|---------|---------|-------------------

there is no reject part of promise in your code, so it is always resolve , only the test cases of successful callback function