Hi Mary, it does look like powdery mildew. Powdery mildew is plant specific, so the one on your day lily is probably not the same mold as on your clematis.

Basic cleaning of old leaves around the plants is necessary.  Plants should receive at least 6 hours of sun. The spores overwinter in old leaves.  Also, improve ventilation in the area to prevent mildew from starting by trimming leaves and maybe transplanting plants.  Water at ground level and don't wet leaves.  Key is to apply what solutions you chose to control the powdery mildew at suggested time intervals all season long. Next year, apply the chemical control agents before the mildew starts so you can control it.  Watch out for leaf health during hot season and spraying, to prevent leaf scorching.  Improve soil aeration around the plants by adding compost.  Don't over fertilize.  

The articles from sources with .edu or .org are research based.  People are using neem oil and even baking soda to help.  I've included a few sources on that.  Baking soda is fungistatic, so it does not kill the mildew.  Solutions using copper and sulfur based products are proven effective.  Always follow the rules about mixing and application of product, along with protective gear for you and washing up after use. UMN Extension article lists products from most effective to least effective.

Generally, plants don't die from powdery mildew but they do not grow well.  I also included an article on day lily care.  Good luck.  Start spraying with the chemical you choose, following all directions, and doing things to increase ventilation and less moisture on leaves and you will help control it this year.  Clean up severely damaged leaves this year may help, using good sanitation.  Always dispose of the infected plant materials-do not compost.  Remove all old vegetation this fall, raking and trimming back vines and leaves.  Do not compost but dispose of infected materials.  They are developing plants that are more mildew resistant but I couldn't find any truly resistant for day lily and clematis. Thank you for your interesting question!

https://extension.psu.edu/clematis-diseases

Powdery Mildew | Home & Garden Information Center (clemson.edu)

Powdery mildew - Outdoors (missouribotanicalgarden.org)

Powdery Mildew on a Clematis (sfgate.com)

Neem Based Insecticides - CT Integrated Pest Management Program (uconn.edu)

Clematis | Home & Garden Information Center (clemson.edu)

Powdery mildew in the flower garden | UMN Extension

Clematis Diseases (psu.edu)

Daylily Diseases & Insect Pests | Home & Garden Information Center (clemson.edu)