My reel mower is pretty bad against broadleaf weeds. A few things to help would be to sharpen the blades and set the reel blades at the proper distance from the cutter bar blade. You do that by using the screws at the ends of the cutter bar. When I first got mine, the distance was set badly and I had to work twice as hard to get a decent cut.

Also, if some particular patch or grass blade isn't getting mowed, I sometimes have to change directions on it or go really fast over it.

Boiling water kills weeds in my driveway, but it's more difficult in the grass as you don't want to kill the grass, too.

You'll probably have to hand dig to remove the weeds, but it's easier after a soaking rain. This time of year might be difficult. A weed hound grips the weed firmly to remove the deep taproot, but the side of your palm gets sore even with gloves.

One spring years ago I paid my 8 year old son per dandelion to finish up the last 100 or so left in the lawn. He eagerly prowled the grass after school & earned a good amount of $ because there were many hundreds still there.

After you remove the weeds you'll need to spread grass seed, but it's easier in September after the rains come again. The weeds come back if you don't fill the space with something else.

We've reduced the size of our lawns and created smooth curves for easy mowing, so is isn't hard to keep up. In fact our teen son mows the front in 7 minutes. I know because he timed himself to prove it's not too much lawn because he wanted us to keep enough turf for lawn games like croquet & bocce ball. On hot days it's nice to lay a blanket on the cool grass in the shade whereas it's not comfy to lay a blanket on wood chip or rocky paths or pavers.

Plus the plentiful clippings in spring are useful in gardens as mulch once dried.

Here is a link that might be useful: weed hound

Way back when--when I was a kid we used a reel mower on our "meadow" lawn. Then in the summer when the plantain stems wouldn't cut we went over that area with a hand weed cutter that looked like a golf club with a long double edge blade at the bottom. You swung it back and forth to chop off those tall stems.

Here is a link that might be useful: grass whip

That's hairy cats-ears you've got there, or summer dandelion. Hypochaeris radicata. Tough to kill. There won't be an easy way to reduce its populations once it's got a hold of your lawn. It loves low-maintenance lawns!

I control it in my lawn by keeping the lawn well watered and actively growing. The hypochaeris is quite drought tolerant and doesn't mind at all going without water - in fact it loves drought and out competes the grass when you let the lawn go dry over the summer. That won't get rid of the ones already in the lawn, though.

It's also fairly tough and fibrous and hard to cut, so yes sharpening the blades, and going at it from different directions will help. I have to do that even with the electric mower.

I've been using 'Ortho Elementals Lawn Weed Killer.' I spot treat the weeds. I'll have to spray the mature plants more than once, but it does kill off the top growth well. I have many types of weeds and too many to remove by hand. There's also a product called Burnout, but it would kill grass too.