Ds was like this for the longest time (from about 4 - 7 months old, then again at 9-10 months old). Even if he saw me he cried. He wanted to be held, however if I was just sitting down this was often not good enough, I had to be doing something.
I took him out as much as I could for walks (pushchair and car were not as good as walking and carrying him).
I also carried him in a sling whenever I could to do things around the house. Some people say you can do this all the time but there were certain thngs I couldn't do with him in a sling so sometimes he was left to cry a bit. It was hell though. All my friends seem to have a baby that was happy to sit and watch them - not mine though lol
At 4 months you could still carry your dd lying down possibly which I found easier when working in the kitchen. Once they sit up they grab stuff so chopping veg etc gets really difficult.
With ds in a sling at home I could vacuum, load/unload dishwasher or wash up, go to the loo, dust, load/unload clothes machine/dryer, hang out washing and sometimes do some preparation for cooking, though once hot water etc was involved he had to go soemewhere safe. You can stir stuff etc, though.
You can't iron (though as kids get to the crawling stage it is safer to do that at nap time and after bed in my opinion)
I would also use the times he was quiet/asleep to do the cooking. I started cooking stuff in advance (sometimes I cooked dinner at 9am when he had a morning nap) and leaving it to be warmed up later. Also if cooking anything - put at least 2 portions in the freezer for those days when she is being extra clingy or you are just tired.
This clingyness goes in phases and ds is recently wanting to be picked up a lot again (14 months). I believe in a middle roead - I do think 40 minutes of leaving her crying is a bit long. 10 minutes or so is sometimes difficult to avoid but if you carry her more you may find life less stressful.
Mind you ds often cried a lot even when being held , particularly if I was tired and sitting down - in that case I just held him and he pretty much cried himself out - on those occasions when I just too knackered to take him out or to do anything.