Learn English – Afternoon OR in the afternoon

In the UK, dinner would normally refer to the main meal of the day, irrespective of the time of day at which it is eaten. It could, for example, be eaten around midday ("lunch time"!), early evening or later evening. The discussion at "Lunch" vs. "dinner" vs. "supper" — times and meanings? already adequately covers that subject.

Tea on the other hand can mean several difference things:

1. It may simply refer to the drink.
2. It may refer to Afternoon tea, which is a particular style of light meal, traditionally eaten at Tea time.
3. It may refer to a main meal, traditionally known as High tea and eaten in the early evening.

If, for example, you were asked Would you like some tea?, it would most likely mean Would you like a drink of tea? but could - depending on the time of day and the situtation - refer to Afternoon tea.

Certainly, one would expect a Tea Shop in the UK (as distinct from a Coffee Shop) to serve traditional Afternoon teas, as well as just serving tea to drink.

Traditionally, Afternoon tea would have been observed by middle and upper classes - and especially by the ladies of those classes. It would consist of tea (to drink) served from a china teapot and drunk from delicate china cups, accompanied by delicate savoury or sweet sandwiches (stereotypically, cucumber sandwiches and jam sandwiches), or scones with jam & cream, and followed by cakes (often homemade).

Some people and families continue to have their main Sunday meal at midday - when it is commonly referred to as Sunday lunch despite being the main meal of the day - and may then have Sunday Afternoon tea as described above. Many may do this more during the winter and/or when having visitors, rather than regularly.

High tea is so-called to distinguish it from Afternoon tea. Traditionally, it was a meal eaten in the early evening, and consisting of a hot dish, followed by cakes and bread, butter and jam. It has tended to be associated more with Northern England and the working class, although sometimes the meal would have been eaten by children of the middle & upper classes, whose parents would have eaten a more formal dinner later in the evening.

There is more information at the Wikipedia entry for Tea (meal), from where some of the above information has been taken.

Progeny is generally used as an uncountable noun.

"My progeny" could refer to 1 child, or to 8 children, 29 grandchildren, and 95 great-grandchildren.

In your two examples, you are trying to use progeny as a countable noun. It's incorrect to do so. "Children" is formal enough to use in your two examples.