No the usual duplex chain has no tensioner fitted.
When they get loose just fit a new one.
But be aware that the chains available now are mostly pretty slack compared to the original Renold ones we got years ago.
The German Iwis chain is good, but you'll pay $50+ for it.
That hole you see is a breather hole, and yes it's normal to have a bit of oil in there. It comes out of a hole in the camshaft retainer plate, behind the big sprocket.
Chains need oilz, unlike timing belts.

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DrMini- 1970 wasaMatic 1360, Mk1S crank, 86.6HP (ATW) =~125 @ crank, 45 Dellorto (38 chokes), RE282 sprint cam, 1.5 rockers, 11.0:1 C/R.
