Something like this...
// Create layer group
var group = L.featureGroup();
// Add layer group to map
map.addLayer(group);
// Add marker at center of map
group.addLayer(L.marker(map.getCenter(),{draggable: true}));
// Setup map 'click' listener to clear existing markers and add new marker at click location
map.on('click', function(e){
group.clearLayers();
group.addLayer(L.marker(e.latlng,{draggable: true}));
});