Remove previous Marker and add new Marker in Google Map v2

Just clear the google map before adding marker. Like this:

@Override
public void onMapLongClick(LatLng latLng) {
    googleMap.clear();

    googleMap.addMarker(new MarkerOptions()
            .position(latLng)
            .title(latLng.toString())
            .icon(BitmapDescriptorFactory.defaultMarker(BitmapDescriptorFactory.HUE_RED)));
}

Just creat a new marker object and before adding a new marker, remove the previous one

Marker marker;

MAP.setOnMapLongClickListener(new GoogleMap.OnMapLongClickListener() {

                @Override
                public void onMapLongClick(LatLng arg0) {
                    if (marker != null) {
                        marker.remove();
                    }
                    marker = MAP.addMarker(new MarkerOptions()
                            .position(
                                    new LatLng(arg0.latitude,
                                            arg0.longitude))
                            .draggable(true).visible(true));
                }
            });

EDIT

Do the same for OnMapClick

MAP.setOnMapClickListener(new OnMapClickListener() {
        @Override
        public void onMapClick(LatLng point) {
            // TODO Auto-generated method stub

                if (marker != null) {
                    marker.remove();
                }
            marker = MAP.addMarker(new MarkerOptions()
                    .position(currentPosition)
                    .snippet(
                            "Lat:" + location.getLatitude() + "Lng:"
                                    + location.getLongitude())
                    .icon(BitmapDescriptorFactory
                            .defaultMarker(BitmapDescriptorFactory.HUE_AZURE))
                    .title("ME"));
            Log.e("lat", "" + point);

        }
    });

Here a simple way You just have to change the position of the marker. Create Global Object as Marker marker;
After that add marker to map like

marker = map.addMarker(markerOptions).position(new Latlng(31.647316, 74.763791));  

And after it use marker.setPosition(newlaLng); where you need to add marker.