No problems with compiling this code here.
Added this to header:
Qt Code: 
  1. #include <QDebug>
  2. #include <QNetworkAccessManager>
  3. #include <QUrl>
  4. #include <QNetworkRequest>
  5. #include <QObject>
  6.  
  7. class ClientHandler : public QObject
  8. {                                                              //    << on this line
  9. Q_OBJECT
  10.   QNetworkAccessManager *manager;
  11. private slots:
  12.   void replyFinished(QNetworkReply *);
  13. public:
  14.   void CheckSite(void);
  15.   ~ClientHandler();
  16. };
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This to .cpp:
Qt Code: 
  1. void ClientHandler::replyFinished(QNetworkReply *reply) { qDebug() << "DONE"; }
  2. ClientHandler::~ClientHandler() {}  // <<-- on this line
  3.  
  4. void ClientHandler::CheckSite(void)
  5. {
  6. QUrl qrl("http://checkip.dyndns.org");
  7. manager = new QNetworkAccessManager(this);
  8. connect(manager, SIGNAL(finished(QNetworkReply*)), this, SLOT(replyFinished(QNetworkReply*)));
  9. manager->get(QNetworkRequest(qrl));
  10. }
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Included header in main.cpp and I can use ClientHandler objects:
Qt Code: 
  1. #include <QApplication>
  2. #include "ClientHandler.h"
  3. int main( int argc, char ** argv ){
  4.     QApplication app(argc,argv);
  5.     ClientHandler h;
  6.     return 0;
  7. }
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Then simply: qmake, make - compiled without errors.
Maybe try to do clean rebuild ( make clean, qmake, make ).