Every time I try to start the engine it takes a minimum of 2 attempts. It turns over fine, and all lights/electrics etc are working but I need to pump the accelerator to get it to ‘catch’. It is usually 2 attempts but occasionally would take 3. and when it does catch it’s a bone-shaker. However this morning it took 5 goes

And when I say ‘every time’ that’s exactly what I mean – cold, warm, wet, dry – it makes no odds. And if I have driven home from work (20 odd miles), stop at Sainsburys for 2 minutes and start again – the same problem, so it’s not just that the engine is cold.
I had the second battery replaced a while ago but that did squat to sort the problem and all electrics are fine - lights mirrors etc. so don't think it can be the electrics
It may be unrelated but on my way up to town last Thursday, having driven for about 40 mins the engine started juddering slightly when I slowed down and shortly after cut out completely when I had to slow down to take a left turn………and on cold mornings it is very sluggish – pressing the accelerator does very little and then suddenly whoosh, she’s off. It also cut out this morning on the garage forecourt but it was first thing........
Conversely, last winter, even in extreme cold, the engine would (start first time &) just rev very high (up at about 2500+ rpm doing 40mph) until it got into its groove and then the revs would drop back to normal. which I understand is totally normal.
anyone??