Right. And oh my do I hate blat.
It's a difficult concept to translate to English because it's not synonymous with corruption or bribes. A one-time bribe transaction isn't blat. You want a school to accept your kid so you "gift" the school some supplies, that's not blat, it's a one-time thing and the school principal doesn't owe you any additional favors. Blat is more like a social network of people trading favors, and each individual transaction within your blat network may involve different things. It could be money, it could be access to a product (that you still have to pay for), it could be time or labor.
Maybe you know a plumber and he will come look at plumbing problems for you and your family, for free or for a low price. But you work at a grocery store and the plumber can always buy cheese because you set some aside for him. That's a blat relationship. And then the blat network grows - one day you mention you'd like to see a theater pay and it turns out the plumber's wife works in a theater and can help you get tickets, he'll set you up. Your husband is an engineer though so he can help tutor their child in mathematics.
I don't know about difficult to translate, sounds a lot like being a "Good Old Boy"
https://en.wiktionary.org/wiki/good_old_boy
https://en.wikipedia.org/wiki/Old_boy_network#United_States
Of course, it has several connotations depending on exact context.
Seems like a big part of it is an extensive barter network since the straightforward exchange of cash for services has broken down, so you need to know the right people with skills or access to, and something of value to offer them in return.
This basically describes how boards of directors and other power structures at that level work. Just with much more expensive assets and favors.
None of those examples you gave sound like corruption to me, with the possible exception of tickets. It seems to me that the problem is when people who are in a position of power and responsibility abuse their power for personal ends. Plumbing or tutoring or cheese are privately held goods and surely the possessors of those goods can dispose of them how they want?
Or perhaps in all of these examples the plumber/grocer/engineer is entrusted with responsibility from the government to ration a scarce resource?