努力工作 用功学习 adjective, verb or adverb
努力 can be a noun for 'great effort' or a verb for 'to strive/ to work hard' 
努力地 (in a hard working manner) is 努力's adverb form 


The sentence "他努力工作"  omitted '地' in  "他努力(地)工作" (he works in a hard working manner)


用功 is also a verb for 'work hard'
用功地 (in a hard working manner) is 用功's adverb form


The sentence "他应该用功学习"  omitted '地' in "他应该用功(地)学习" (he should study in a hard working manner)

Examples of 努力; 用功 as verb:

"他在努力" - "He is working hard"

"不要阻他用功" - "Don't stop him from working/ study hard"

Most common usage of 用功 is  "to study hard"

Edit: 


  can you omit anytime with any adjective you use as an adverb?


It depend on the context


強烈 (strong)
強烈地 (strongly) 


Look at the following two sentences:


"強烈反對" = [強烈(adj)+反對(n)] = "strong objection" 
"強烈地反對" = [強烈地(adv)+反對(v)] =  "strongly object"


If you omit 地 in sentence #2, it would become the same as sentence #1 . We would not know "強力反對" mean "strong objection" or  "strongly object". 

But with enough context, we can tell which is which.

"(他)強烈反對" = "(He) strongly objects" - With the subject "he", we know 強烈反對 must be [adverb +verb] . Therefore, we can omit the adverb marker '地' in "(他)強烈(地)反對"

"(他的)強烈反對" = "(his) strong objection" - With the possessive pronoun "his" , we know 強烈反對 must be [adj + noun]