The problem here is that you're trying to get a plural sense out of an uncountable noun. We often have alternative ways about talking about things that are normally uncountable in a countable way. So the following are usually uncountable in English:
We we want to talk about them in the plural or singular we can often make these words parts of larger phrases. So we see:
- a piece of information
- three glasses of milk
- two pieces of paper
With news the easiest way to do this would be to use the compound noun news stories. We can use either this kind of with a singular noun, or these kinds of with plural or singular countable nouns:
- This kind of car
- These kinds of car
- These kinds of cars
However we can never say:
The word this always goes with the singular noun kind, and these always goes with the plural, kinds.
The Original Poster's best option, therefore, is probably:
- I have been coming across these kinds of news stories lately ...
Hope this is helpful!