此问题可能是由于API网关未正确配置所致。以下是一个示例配置，可确保向所有客户端发送Websockets消息：

{
  "Type": "AWS::ApiGatewayV2::Api",
  "Properties": {
    "ProtocolType": "WEBSOCKET",
    "Name": "example-api-websocket",
    "RouteSelectionExpression": "$request.body.action",
    "Tags": {
      "Name": "example-api-websocket"
    },
    "CorsConfiguration": {
      "AllowCredentials": true,
      "AllowHeaders": [
        "Content-Type"
      ],
      "AllowMethods": [
        "GET",
        "POST"
      ],
      "AllowOrigins": [
        "*"
      ]
    },
    "RouteSettings": {
      "DataTraceEnabled": true,
      "DetailedMetricsEnabled": true
    }
  }
}

确保正确设置了“RouteSelectionExpression”，以便在发送消息时选择正确的路由。另外，检查在调用API时是否正确发送了正确的参数和格式化的负载。