How to use this photo to estimate the distance from the Earth to the Moon during the annular solar eclipse on June 21, 2020. We only know that the distance from the Earth to the Sun that day was 152036000 km. Please help me with this difficult task - I've been working on it for a week, but I can't find anything worthwhile on the Internet. 

Sun diameter = 1.392x10^6 km

Moon diameter = 3475 km

Earth to Moon distance = 152036000 / 1.392x10^6 * 3475 = 380000 km

It’s not enough information. The moon could be large and close to the sun, or small and close to earth, or anywhere in between. You just know that the angular size is about the same.

If you knew the exact time and date of the picture, and where it was taken.
And you also had a second similar picture from elsewhere, and knew the time  date etc for that one, I think you could triangulate the position of the moon by assuming the sun is "very far away".

But I think that's doing it the hard way.
 

If I understand the task correctly... it is to estimate the moon distance on that exact day (it should be somewhere between 362600 and 405400km). You can take the moon and sun diameter (both fixed) from any external source.... The interesting part, imo, would be to estimate the error of your calculation.

Here is another way. The distance to the Sun and its angular size let you find its diameter. Now you have two unknowns, the distance to the Moon and its diameter. You need two equations. The picture gives you one equation based on the equal angular sizes of the Sun and the Moon.

If you can measure the time between the Moon touching the Sun and the moment in the picture, it gives you time that takes the Moon to move the distance equal to its diameter. We know how long it takes the Moon to move the entire length of its orbit around the Earth. Comparing these two times gives you the second equation. Then you solve two equations with two unknowns.