When I push the front wheel, it has some play. Also, when I brake at low speed, I can feel some steering wheel vibration and a knock, can this indicate that the "lollipop" needs to be replaced?
Could be a number of things including as you say a front lower lollipop arm but not sure that would knock. I would look at drop links which are fairly easy to replace or anti roll bar bush or of course play in the steering arm itself. Think I would get a garage to diagnose although these things take a while to develop so would be picked up as advisories on MOT.