How to correctly use the API routes in the Next JS 13 Using App Router framework

I'm fairly new to React JS and Next.js. I was previously using the Page Router API in Next.js but then switched to use the new App Router in Next.js 13.

Previously, using the Page Router, the structure to create a single GET request was to nest your "JS" file under the page\api. The server automatically creates the route and returns the methods + data. For example, I would export the following function:

export default (req, res) => {
  res.statusCode = 200
  res.json({ name: 'John Doe' })
}

Now, using the App Router and TS on Next.js 13, I understand that the mechanism is different. The API folder is nested under the app folder and you have to define a "route.ts" file with all the different GET, POST (etc..) methods:

import { NextApiRequest } from 'next';
import React from 'react';


export async function GET (request:NextApiRequest){
    return new Response('John Doe')
} 

I have some problems understanding how to use this Response object. How to you specify the structure of what you return, in my case { name: 'John Doe' } ? I want to change the response status to 400, how do I do that?

Going forward, I have to deal with a POST request.

import type {NextApiRequest, NextApiResponse} from 'next'

type MyData ={
    name:string
}

export async function POST(
    req: NextApiRequest, 
    res: NextApiResponse<Data>
){
    
    const {nameLookup} = req.body
    if(!nameLookup){
        
        //WHAT I WOULD LIKE TO DO IF I WAS USING THE PAGE ROUTER 
        //BUT THIS IS NOT WORKING WITH THE APP ROUTER

        res.statusCode = 400
        //- error TypeError: res.json is not a function
        res.json({name:'Please provide something to search for'})
        return res
    }

     //WHAT I WOULD LIKE TO DO IF I WAS USING THE PAGE ROUTER 
     //BUT THIS IS NOT WORKING WITH THE APP ROUTER
     //- error TypeError: res.status is not a function
     return res.status(200).json({anwser:'John Doe'})
})

Thanks for helping, the learning curve is a bit steep having to deal with everything at the same time (TS, Next.js 13)

Here is an example based on your route:

export async function POST(request: NextRequest) {
  const { nameLookup }: MyData = await request.json();

  if (!nameLookup) {
    return new NextResponse(
      JSON.stringify({ name: "Please provide something to search for" }),
      { status: 400 }
    );
  }
  
  return new NextResponse(JSON.stringify({ answer: "John Doe" }), {
    status: 200,
  });
}

Please not that you should do some research on your own before posting to stack overflow. A look at the app router documentation here would've probably sufficed.