The word صَباحاً does not mean in the morning (as in a fixed time [words in Arabic have meaning and application]). Instead of صَباحاً, it would be most appropriate to say في الصباح. This is because the word في means "in" and the letters ال together mean "the". So, all together (including صباح), it means "in the morning".
The word صَباحاً is normally used in time. For example, we would say ًالعاشرة صباحا, which means "ten in the morning". The word صَباحاً is a حال (adverb). So, it is not appropriate to say the word صَباحاً in the sentence.The word صَباحاً does mean "in the morning" but in the sense of time only.
Also the word اَلْأَكْل in the sentence above can be confused with the word "food" (informal) because of the missing حركة (accent) above the letter ل. The word should be اَلْأَكْلَ. The sentence should say " في الصباح، أُحِبّ اَلْأَكْلَ هنا" which means "In the morning, I like eating here".
I agree with you about the missing vowel (حركة). I also agree regarding that صباحا is used when you specify a time. For example "I will run seven 'clock in the morning": سَأَركُضُ السَاعَةَ السَابِعَةَ صَبَاحَا
Except for telling time, I have heard that صباحا and في الصباح are synonyms (but of course like most synonyms there are small differences). That is also what FIX says in a comment above, and what my friend from Syria said when I recently asked him. Therefore I am quite sure you could choose between في الصباح and صباحا in this sentence. Just like these two sentences both mean "I love running in the morning":
أُُحِبُّ الرِكضَ فِي الصَبَاحِ
أُحِبُّ الرِكضَ صَبَاحاً
صَبَاحاً is an adverb of time Adverb of time is called ظرف الزمان in Arabic.
I have never heard the term حال for adverb in Arabic. There is a term called الحال that is used to described the state of the subject of object at the time when the action of the verb took place. الحال also has accusative state
You are right in that في الصباح literally means "in the morning". The word صباحا literally means "a morning" and is in accusative state. It is very interesting and fun to analyze texts literally. But of course, just because something can be translated to something literally, it does not mean that it is the correct translation.
I did not say it was not valid. I said it was not appropriate. I speak Arabic fluently and I have never heard صباحا be used outside of the tense/time section. It can be referred to as valid. However, it is quite unnatural to use this word in this sentence. I said صباحا is used for time as I could not find any "appropriate" scenario for this word. Also, I did some research and all the examples I found online were to do with time and 'date' (as in tomorrow or so). I hope that clarifies it.
https://youtu.be/bh4p1VtFmJs?list=PLuvzovgAD3aqMdLfL2TKUbuqNVgJd6W_A&t=150 Here it says that every noun in accusative that indicates the time when the verb happened, is called ظرف الزمان This is also what I learned in university. Interesting that there are so many different theories of Arabic grammar. But also logic since it is a widespread language with dialects and MSA and classical Arabic. The differences started almost a thousand years ago with the school of Basra and the school of Kufa.
Thank you. Some people say that Arabic grammar is totally different from Indo-European. But to my ignorant eye there seem to be a lot of similarities - names of parts of speech, eg adverb, noun... cases (though only three as far as I can gather - nominative, accusative, genitive), articles (though only one, so far - ال. I've probably got some horrible incomprehensible surprise awaiting me...