Tl; dr: We don't have any decisive contemporary account, but the majority of recent scholarship (including myself) are negative in this question (Körner 1964; Walker 1997: 154f.)
The only possible "legitimate" claim that Harald could have (and some later traditions cite) is the alleged treaty between King Magnus the Good (d. 1047) of Norway (nephew of King Harald hardrada) and King Hardeknud (Harthacnut) of Denmark (son of King Cnut the Great) concluded in 1037/38:
"Is it true what I have heard people say in England, that your kinsman King Magnús sent men to King Eaðvarðr, and it was in the message that King Magnús had inherited England just like Denmark from Hǫrða-Knútr, as stood in their oaths?" (Haralds Saga Sigurðarsonar, Chap. 79, in Heimskringla. The translation is taken from: [Finlay & Faulkes trans. 2015: 105]).
The alleged existence of this treaty as well as the oath that acknowledged the outlived one of the two rulers to inherit another throne has mentioned both in Denmark and in Norway since about the middle of the 12th century (as for the text, please check my previous post in: Was the peace Treaty between Cnut the Great and Edmund Ironside unusual?). Some historians have disputed the accuracy of the relevant sources, I suppose that the agreement itself, supported by the participants (magnates) of both parties, might have actually been concluded.
Nevertheless, there is an clear leap in logic between the treaty mentioned in those preceding text and that suggested above in the 13th century Heimskringla:
"King Magnús won Denmark because the leading men of that land helped him, and you did not get it because all the people of the country stood against you. King Magnús did not fight to conquer England because the people of the country wanted to have Eatvarðr as king. If you want to gain England, then I can bring it about that the majority of the leaders in England will be your friends and supporters." (Haralds Saga Sigurðarsonar, Chap. 79, in Heimskringla. The translation is taken from: [Finlay & Faulkes trans. 2015: 105f]).
As I explained before in: What was Sweyn Forkbeard's claim on England's crown, and was it at all legitimate by the standards of the day, or was it something of a simple overthrowing takeover?, the conquest plus local recognition strategy had actually been employed by Sweyn Forkbeard and his son Cnut, and ended in success. Harald's misjudgement was probably the over-optimistic estimation of the local support found in northern England, or just to overestimate Tostig's popularity in England.
On the other hand, Harald's expedition should not certainly been identified with the stereotyped sporadic raiding of the Vikings in the 9th century. The fleet consisting of 300+ ships would nearly correspond with the kingdom-wise mobilization (The 14th century law book stipulated that 336 ships were to be mobilized in the total kingdoms, based on the naval conscription of the farmers). If we believe later saga writing, Harald also brought his female royal family (Queen and daughters) up to the Orkney Isles in the northern fringe of the British Isles (Sveaas-Andersen 1977: 166). Sveaas-Andersen proposes that Harald might planned to extend his political influence in the northern part of the British Isles first on the Orkney Isles, then on York where he met stronger local resistance as well as Harold's army from the south (Ibid.: 166f.).
If you are interested further in the political chaos around the North Sea between 1042 to 1066 (and possible after), the following post by me might also be interesting: Anglo-Saxon Historians - Would the Godwin / Godwinson dynasty be able to call upon political support / military support coming from Denmark ?
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