You've picked a tough time to start fishing a lake.
The big question is are you sure you have found the fish?
Sounds obvious I know but no point fishing the margins if they are in the middle of the lake.
Spots to look for the fish, near snags, tree roots, maybe any rushes, dead or dying weedbeds, possibly natural food beds (bloodworm), which can be found by casting out and dragging a lead along, the grippa leads are good for this as
@ouchthathurt mentioned.
Over leaf litter is when I pretty much stick exclusively to pop-ups, making sure I have my bait above it, although in silt I try to get my bait to the level the fish are feeding at.
That doesn't sound like it makes sense, but carp might not feed on top of the silt, but dig into it, picking up mouthfuls of natural food. On other lakes they might feed on top of the silt, especially if its horrible stinky stuff.
Talking of stinky stuff, leaf litter I've heard can make the area more acidic, and put the fish off feeding for a while.
On my syndicate lake, the lake is a pretty consistent depth, no drop-offs other than the old stream bed, however there are some softer holes in the silt where carp have dug them out. Imagine the lake is 4feet deep, and then you suddenly drop to 5feet deep. These holes gradually fill back up, but the bottom is softer.
Those pop-ups, about the only time that I have a bait off the lake bed, normally I fish close to the lakebed, the counter weight directly below the hook, but in leaf litter I will fish with the weight a thumb or even forefinger length below the hook.
Are you able to attract smaller silver fish and get them feeding? Their feeding might actually get the carp investigating.
For that a groundbait with very little solid food content, maybe fine ground breadcrumb, liquidised or creamed sweetcorn.
Hookbaits, obviously try sweetcorn, maybe maggots or boilies.
Boilies: you might find in winter certain flavours work better than others. I used to do very well on Dynamite Green Zing, Solar and Nutrabaits Pineapple and N-butyric pop-ups, along with my own recipe Garlic and Spice ones.
The occasional fish in winter on my syndicate comes out on pink pop-ups.
I have to keep going back to it, years ago, I did a session between Christmas and New Year. For 2 days I caught on the high attract pop-ups mentioned above (my original garlic spice) then after feeding a few of my normal food bait boilies every day, I started catching on them, ordinary bottom baits. In that week I think I ended up with 10fish, nothing massive, but any fish in freezing winds and temperatures near 2 or 3⁰ Celsius is a good fish.
The yellow and green pop-ups on the second lake worked on day trips when we hadn't got a bait established.
On those two waters Other pop-ups just didn't cut it!
Braid definitely gives a better sense of feel than mono.