...
void myWidget::on_pushButton_clicked()
{
...
timer->stop();
timer->setInterval(8000);
timer->start();
}

This code is a bit reduntant. simply calling timer->start(8000) will pretty much do the same;

Here is a quote from Docs (link):

Starts or restarts the timer with the timeout specified in interval.
If the timer is already running, it will be stopped and restarted.
If singleShot is true, the timer will be activated only once.

myWidget::myWidget()
{
timer = new QTimer(this);
m_Button = new QPushButton("Click");
widget = new QWidget;
timer->setSingleShot(true);
connect(m_Button,SIGNAL(clicked()), this,SLOT(on_pushButton_clicked()));
connect(timer,SIGNAL(timeout()), this,SLOT(on_TimerTick()));
}
void myWidget::on_TimerTick()
{
widget ->hide();
}
void myWidget::on_pushButton_clicked()
{
widget ->show();
timer->stop();
timer->setInterval(8000);
timer->start();
}

QTimer *timer = new QTimer(this);

And in your "trigger" function you should do something like this:

timer->setSingleShot(true);
timer->start(timeout); //If timer was running, it will restart

Once the pushbutton is clicked, show the widget, if the timer is already active, stop it, set the interval to 8 seconds and then start the timer. Once after the timeout, hide the widget.