Question: Are 'hayaan' and 'payagan' completely interchangeable?

@kuya19 non-native speaker here, but I get the impression they are different flavors of “allowing,” in that “payagan” takes on a connotation of allowing as in “consenting” or “acquiescing”, whereas “hayaan” takes on a connotation of “not impeding” something.

@kuya19 Let’s fix your example sentences first. We need the linker “na” for “huwag hayaan/payagan -> huwag NA hayaan/payagan” and “hayaan/payagan gawin -> hayaan/payagan NA gawin”. The sentences should therefore be:Huwag moNG hayaan ang mga bata moNG gawin ‘yan.
Huwag moNG payagan ang mga bata moNG gawin ‘yan.
The answer given by @jkos is correct. The idea of “allowing” is common to both, but strictly speaking, they are not interchangeable because “hayaan (let it be)” is more about an event that has already begun while “payagan (give permission)” is more about an event that has yet to begin. The event under “hayaan” could be ongoing at that moment (present tense) or a series of discrete, but identical, events that began in the past and are still expected to happen at present (present perfect progressive tense). Its relation to the “present” should be reflected in the verb it refers to such that the first sentence should have been:Huwag mong HAYAAN ang mga bata mong GINAGAWA ‘yan. (They are doing it right now/They have also been doing it in the past.)Conversely, since the event of “payagan” is yet to happen, the verb should be in the infinitive.Huwag mong PAYAGAN ang mga bata mong GAWIN ‘yan. - Your example is correct.We have a tendency though to use them interchangeably to mean “allow” and we just do the mental adjustments to get the intended meaning. Your two examples, with the “na” added, therefore, would actually sound just fine to us. In fact, even “Huwag mong PAYAGAN ang mga bata mong GINAGAWA ‘yan” is something you might hear some of us say.Other variations: 1. We might also use “hinahayaan” or “pinapayagan” instead, which suggests that the person being told so has been allowing them to happen in the past. Though that assumption might be incorrect, it would still be understood in its proper context. 2. At times “bayaan” or “pabayaan” is used instead of “hayaan”.Ergo, just be aware of the grammar rule, but don’t worry about deviating from it since we native speakers occasionally do it, too. 😅Finally, just in case you used “mga bata” in your examples to mean “children”, that meaning changes when you add “mo” after it. In your given sentences, “mga bata mo” would be understood as “your underlings”. If you wanted to say “your children” it has to be “mga anak mo”.

@kuya19 Let me improve your sentences. (Hayaan mong ayusin ko ang iyong mga pangungusap.)
1. Huwag mong hayaang gawin iyan ng mga bata. (O kaya "Huwag mong hayaang gawin iyan ng iyong mga anak.") - Do not let the children do it. (Or, "Do not let your children to do it.
2. Huwag mong payagang gawin iyan ng mga bata. (O kaya, "Huwag mong payagang gawin iyan ng iyong mga anak.) - Do not permit the children to do it, (Or, Do not permit your children to do it.

No they are not.Hayaan is more of like "let it be"/don't interferePayagan is explicitly giving permission

@kuya19 This is just to explain further the meaning of “mga bata mo” and its appropriate usage. I mentioned above that when “mo” is added after “mga bata” its meaning becomes “your underlings”. The same is true with adding “ko” and “niya” after it. When their corresponding plural pronouns are used it is the context that will determine if it should be understood as “children” or “underlings”.However, the use of “mga bata ko/mo/niya” as “my/your/his/her underlings” is not standard practice and is more of street talk. The standard one to use in place of “bata” is “tauhan (employees/staff)”.The singular noun “bata” also has its slang meaning of “girlfriend” or, less often, “boyfriend”. The younger generation, beginning with the Millenials, practically don’t use it anymore as it has already been upstaged by “syota” and “jowa/dyowa”.

Thank you! If you have time, can you also explain why anak and bata both mean kid but it's weird to say 'anak mo'? Like how did 'bata mo' end up as 'underlings'? Is it simply just how it's used?

@kuya19 No, it's not weird as long as it means "someone's child"."Bata" can mean "child", "young boy/girl", or "young" in Tagalog. But for "child" we use it only to mean "anyone's child". Once the parent of the child is identified, we use "anak". The child is playing outside. = Naglalaro ang bata sa labas.My child is playing outside. = Naglalaro ang anak ko sa labas.
Your child is playing outside. = ... ang anak mo ...
His/Her child ... = ... ang anak niya ...
Their child... = ... ang anak nila ...
Peter's child ... = ... ang anak ni Peter ...
Your cousin's child ... = ... ang anak ng pinsan mo ...
Our neighbor's child ... = ... ang anak ng kapitbahay natin ... So "bata mo" has to mean something else. The "underling" meaning suggests "one considered as a "child", an inferior, to/by someone". In Tagalog movies with cops and gangsters, you might hear it used to mean a subordinate or one they have power over.