The first thing people look for soon after they buy their phone is mobile phone cover No need for "a"

___PART_1___ I would write this sentence like this: “The first thing people are looking (1) for (2), (3) shortly after they have bought (4) a (5) mobile phone (6), (3) is a case (7) for it (6, 8).” (1) I would use the present progressive tense here, because the act of looking for a cover is not simply like: “Look once and you’re done”; but something extended and repeated, and therefore the present progressive is well suitable, if not even mandatory, in this case. (2) “To look for” is not separable; so, a sentence like “I look a cover for.” makes no sense. Instead it always has to be: “I look for a cover.”. With another verb and another phrasal construction the separation of the preposition suddenly becomes possible: “The mobile phone which I am searching a cover for is a Wiko Jerry.”. My favourite English teacher on YouTube, Aly (see the channel: “Papa Teach Me”), would probably say now: “Yes, English is ridiculous!...”. (3) And now to all of you reading this: How often did it happen to you that you’ve read a sentence and read and read it on and on, and suddenly the words became strange, and seemingly the whole sentence made no sense anymore… The explanation for this is simply that you have read across the end of an inserted subordinate clause. Therefore commas are recommended around the subclause [1]. Use them, and your readers will be thankful to you!

I'm American. We would say "a" phone cover. "For" is misplaced; it should be a collocation with "look". Technically, "their" should be followed by a plural noun. However, these days people commonly say "their" with a singular noun to avoid the awkwardness of "his or her" or favoring one sex over the other. Language is constantly evolving. https://grammarist.com/usage/they/ "The first thing people look for soon after they buy their phones is a mobile phone cover/case."

___PART_2___ (4) The signal word “after” already shows it: One action is finished and another one is immediately going off;.so, it is not only natural but strongly advised to use the present perfect tense here. [2] (5) “their” looks completely odd and awkward here. Instead use “a”. (6) Instead of boring the reader: use the term mobile phone only once and then refer to it by a pronoun. (7) Most people seem to use “case”, not “cover”. (8) “ to it” could be left out, here. But with it the sentence looks nicer to me. Why? An itch in the elbow tells me that... ;-) But by using another gerund the elegance of the sentence can be further enhanced: “The first thing people are looking for, shortly after buying a mobile phone, is a case for it.” Why this is better? Ask my elbow! And by the way: Is this question a quiz; or why are all the other answers so far beyond all reason? [1] https://grammar.collinsdictionary.com/easy-learning/the-comma [2] https://www.englisch-hilfen.de/en/grammar/present_perfect_simple_past_contrasted.htm

Sorry, I'm going to have disagree with Nanren (no offence, I'm a kiwi myself). I still don't think that you need the word "a". If you were saying "after they buy their phone is an insurance policy" than yes, but "mobile phone cover" is more generalised and therefore the "a" isn't required.