A voltmeter will not necessarily show up poor contact on battery connections, unless testing is done under load.
By HT leads do you mean the battery to starter leads? I normally read HT to mean "High Tension" as in 15,000 volts or so through spark plug leads.
If there is a poor contact somewhere in your starter wiring, you might still show full voltage with the tester and could have a trickle of amperage flow through, enough to make the click, but that is all.
For example, I recently had an oxidized battery terminal connection on a tractor. With key on enough current could flow through the oxidized connection to allow all of the low draw accessories to work, tailight, sprayer controls etc. Turning on the headlights was too much tho, this would shut off everything else. The starter system got just enough current to make the solenoid click, no more.
Available Voltage everywhere read normal, but not under load of course.
A good cleaning of the terminal and lug made everything work as should, starter included.
So if you have not already done so, take all connections in the starter circuit apart and clean them, a brass brush works well but any wire brush will do. Even if they look relatively clean already they may have a layer of oxidation impeding the flow.
Glen
