The plant is indeed a hellebore (not native) and the problem is aphids. Hellebores have their very own aphid species (Macrosiphon hellebori) and the white is just discarded skin from the maturing insects. These typically do not bother the plant until the flowers mature, seed capsules form and the flowers start to go over. You can simply cut back the flowering stems and hose down the plant with a strong stream of water or spray with insecticidal soap or neem oil.

Looks to me like vertebrate pests; my money is on birds. They tend to be attracted to visible ripening fruit, and stab it with their beaks. I get very similar marks on my nectarines. Rats will sometimes get to ripening fruit which is more buried in the plant, but they leave distinct small tooth marks and ragged edges, not these neat slashes. You can protect the fruit by putting a cover over the fruit so the birds don't see it; paper cones or baseball caps (fix the back strap around the stem) seem to work well. More concealed fruit should be safer. --Alison

Blossom end rot. Info is in the forum's faq.

Hi, Our tomatoes and potatoes have this and even some on the cukes. There is also white spots just under the skin (brown spots on the potatoes). I had a soil test done last spring and it shows a very low pottasium level. I wonder if that could be the problem? Our melons also rot before they get ripe. Help- ozarkjean