Here we have three verbs: promise, try and heal. After "promise", we use the infinitive. After "try" (with the meaning of attempt), we also use the infinitive, so we end up with two infinitives in one sentence. A more usual, day-to-day sentence could be "I promise to try to help you later" or "He promised to try to do the work that evening". I hope that's helpful!

Check out collocations and verbs followed by gerunds.

The text is correct and natural. In this case, the verbs promise and try take infinitive objects. The online Collins dictionary for learners is an excellent resource. We see that promise can take the form [VERB to-infinitive], that try can take the form [VERB to-infinitive], and that heal can take the form [VERB noun]. - - - - promise https://www.collinsdictionary.com/dictionary/english/promise [VERB to-infinitive] 1. VERB If you promise that you will do something, you say to someone that you will definitely do it. The post office has promised to resume first class mail delivery to the area on Friday. [VERB to-infinitive] - - - - try https://www.collinsdictionary.com/dictionary/english/try [VERB to-infinitive] 1. VERB If you try to do something, you want to do it, and you take action which you hope will help you to do it. He secretly tried to block her advancement in the Party. [VERB to-infinitive] - - - - heal https://www.collinsdictionary.com/dictionary/english/heal [VERB noun] 1. VERB When a broken bone or other injury heals or when something heals it, it becomes healthy and normal again. No doctor has ever healed a broken bone: he or she sets them. [VERB noun]

Yes, "promising to try to heal a divided nation" is being used as a complex adverb phrase defining the verb "to preside." This kind of style is often seen in news stories. It's not a construction that would flow naturally in conversational speech, but it's certainly not incorrect. To make it a little more conversational, you could break it into two simpler sentences: He is promising to try to heal a divided nation. He will preside over a diverse and fragile coalition that has eight parties with deep ideological differences.