Which verses from scriptures describe hanuman as Sankata Mochana/mochaka, remover of dangers or calamities or difficulties?
This is how Hanumanji starts his statement in Sarga 2 of Kishkindha Kand of Valmiki Ramayana:
ततः तु भय संत्रस्तम् वालि किल्बिष शन्कितम् |
उवाच हनुमान् वाक्यम् सुग्रीवम् वाक्य कोविदः || ४-२-१३
संभ्रमः त्यजताम् एष सर्वैः वालि कृते महान् |
मलयोऽयम् गिरिवरो भयम् न इह अस्ति वालिनः || ४-२-१४
Then Hanuma, the ablest sentence-maker, said this sentence to Sugreeva who is dismayed with fear doubting Vali's mischief.
Perplexity be warded off... all of you discarded that great fear from Vali or his cruelty... there is no scare from Vali here on this best mountain, for it is Mt. Malaya...
This opening assurance by Hanuma to 'ward off any perplexity...' may be noticed. These statements and deeds are befitting to the fame of Hanuma to ward off any evil force.
Sankatmochan Hanumanashtak (संकटमोचन हनुमानाष्टक) describes Hanuman as Sankatmochan:
Few verse are as below:
बाल समय रबि भक्षि लियो तब तीनहुँ लोक भयो अँधियारो।
ताहि सों त्रास भयो जग को यह संकट काहु सों जात न टारो।
देवन आनि करी बिनती तब छाँड़ि दियो रबि कष्ट निवारो।
को नहिं जानत है जग में कपि संकटमोचन नाम तिहारो॥१॥
In the very childhood, when you O Hanuman!, devoured the rising Sun, darkness prevailed all throughout the universe. Consequently, the entire world got frightened and felt that the calamity could not be dispelled by anybody. When gods appeared and prayed, you released the Sun, thereby dispelling the distress of the world. O Hanuman! Who does not know in the world that you name dispels all stress!
बालि की त्रास कपीस बसै गिरि जात महाप्रभु पंथ निहारो।
चौंकि महा मुनि साप दियो तब चाहिय कौन बिचार बिचारो।
कै द्विज रूप लिवाय महाप्रभु सो तुम दास के सोक निवारो।
को नहिं जानत है जग में कपि संकटमोचन नाम तिहारो॥२॥
Frightened by Bali, Sugreeva was living on top of the hill, to which the former had no access because of the curse of a sage. One day not knowing that it was Lord Rama who was approaching the hill with his brother Laxmana, Sugreeva was perturbed and dispatched you O Hanuman, to ascertain who they were. Going to them in the guise of a Brahmin, you brought them to Sugreeva and helped Lord Rama rid Sugreeva of the trouble. O Hanuman! Who does not know in the world that you name dispels all stress!
अंगद के सँग लेन गये सिय खोज कपीस यह बैन उचारो।
जीवत ना बचिहौ हम सो जु बिना सुधि लाए इहाँ पगु धारो।
हेरि थके तट सिंधु सबै तब लाय सिया-सुधि प्रान उबारो।
को नहिं जानत है जग में कपि संकटमोचन नाम तिहारो॥३॥
When Sugreeva commanded the monkeys, led by Angad to go in search of Sita, he warned them that the life of none would be spared, in case they returned without any news of Sita. When the party got tired after an intense search and was stranded by the seashore, you summoned up courage and flew to Lanka and brought news about Sita after meeting her, thus saving the lives of your companions who were immersed in despondency. O Hanuman! Who does not know in the world that you name dispels all stress!
रावन त्रास दई सिय को सब राक्षसि सों कहि सोक निवारो।
ताहि समय हनुमान महाप्रभु जाय महा रजनीचर मारो।
चाहत सीय असोक सों आगि सु दै प्रभु मुद्रिका सोक निवारो।
को नहिं जानत है जग में कपि संकटमोचन नाम तिहारो॥४॥
Instructed by Ravana, when demonesses were troubling Sita in Ashokvana, it was you who through a dream made the demonness Trijata restrain others from doing so, by telling that Ravana’s end was near. Then you also killed a big demon. On account of Ravana’s harassment, when Sita asked the Ashoka tree to give her some fire for ending her life, it was you O Hanuman who dispelled her grief by giving her the ring of Lord Rama. O Hanuman! Who does not know in the world that you name dispels all stress!
बान लग्यो उर लछिमन के तब प्रान तजे सुत रावन मारो।
लै गृह बैद्य सुषेन समेत तबै गिरि द्रोन सु बीर उपारो।
आनि सजीवन हाथ दई तब लछिमन के तुम प्रान उबारो।
को नहिं जानत है जग में कपि संकटमोचन नाम तिहारो॥५॥
During the battle in Lanka, when Laxmana was struck by a deadly arrow discharged by Meghnaad, the son of Ravana and was lying unconscious, almost on the verge of death, it was you O Hanuman who brought the divine Physician Sushhena from Lanka, procured the life-saving herb ‘sanjivani’ from Drona Hill to revive Laxmana back to life. O Hanuman! Who does not know in the world that you name dispels all stress!
Few of the above events are verified in this chapter of UTTARA KANDA of Valmiki Ramayana. Few other events are verified in other KANDAs of Valmiki Ramayana such as Kishkindha, Sundara and Yuddha.
Sankatmochan Hanumanashtak (संकटमोचन हनुमानाष्टक), literally Eight verses for Hanuman, the Remover of Afflictions, is an Awadhi work of eight verses in the Mattagajendra metre, devoted to Hanuman. It is believed to have been composed by Tulsidas on the occasion of the founding of the Sankatmochan Temple in Varanasi. The work is usually published along with Hanuman Chalisa.