AWS CDK目前不支持在API Gateway集成响应定义中直接设置Content-Type。但是，我们可以通过在Lambda函数中手动设置响应头来达到相同的效果。以下是一个示例Lambda函数，它返回JSON响应并设置Content-Type头：

import { APIGatewayProxyHandler } from 'aws-lambda';

export const handler: APIGatewayProxyHandler = async (event) => {
  const responseBody = {
    message: "Hello, world!"
  };

  return {
    statusCode: 200,
    headers: {
      "Content-Type": "application/json"
    },
    body: JSON.stringify(responseBody)
  };
};

在CDK中，我们可以使用lambdaIntegration帮助程序包装Lambda函数，并将其集成到API Gateway中：

import * as apigw from '@aws-cdk/aws-apigateway';
import * as lambda from '@aws-cdk/aws-lambda';

// Define the Lambda function
const handler = new lambda.Function(this, 'MyFunction', {
  runtime: lambda.Runtime.NODEJS_14_X,
  code: lambda.Code.fromAsset('path/to/my/function/code'),
  handler: 'index.handler',
});

// Create the API Gateway REST API
const restApi = new apigw.RestApi(this, 'MyRestApi');

// Define the API Gateway integration for the Lambda function
const lambdaIntegration = new apigw.LambdaIntegration(handler);

// Define the API Gateway resource for our endpoint
const resource = restApi.root.addResource('hello');

// Define the API Gateway method for our endpoint
resource.addMethod('GET', lambdaIntegration, {
  // Define any API Gateway method options, such as request / response parameters
  // ...
});

// Set the API Gateway deployment options, such as stage name and description
// ...

通过这种方法，我们可以手动设置Lambda函数的响应头，并将其与API Gateway集成。