Yes you can.
If you are adding a new widget you can choose "My Location" from the widget settings that pop up immediately. for the new widget location and that widget should then follow your location.
If you have already placed a 1Weather widget for a city, you can change the location by going into the widget settings. Open 1Weather>Menu>Settings>Widgets to access the widget settings. Now tap the widget you want to customize and select the 'location' option and choose 'My location'.
Note that you should give location permissions to the app to enable this feature.