Both are possible. In order to decide which tense is appropriate, we first have to distinguish the use of the simple preterite (or simple past tense) and that of the preterite perfect (or past perfect tense).
The past perfect tense is used to locate the situation anterior to an intermediate time which in turn is anterior to the time of speaking/writing (i.e. it's doubly anterior). For example, if I say to you "By the time I arrived, Ed had gone home", the time of my arrival is anterior to the time of my utterance, and the time of Ed's going home is anterior to my arrival, i.e. one stage further back in time, hence doubly anterior. Thus the simple preterite "By the time I arrived, Ed went home" would be incorrect (and ungrammatical)
By contrast, the simple preterite locates the entire situation anterior to the time of speaking/writing. So if I say to you "I was at Ed's place and Kim made the tea", it means that my visit to Ed was anterior to the time of my utterance, and Kim's tea-making was happening simultaneously, while I was there, not at some previous (anterior) time, so the doubly-anterior past perfect "I was at Ed's place and Kim had made the tea" would be wrong.
Bearing those two uses in mind, let's see which tense best fits your example. The statement "It was influenced by the shopping malls" tells us that the time of the influencing was anterior to the time of speaking/writing, though we don't know exactly when (it could have been in the 1950's or later). Now consider the sequence "began to emerge in the 1950's". If we assume the emergence to have occurred prior (anterior) to the time of the influencing, then the preterite perfect ("had begun") would be appropriate. But if the two situations (the influencing and the emergence occurred at roughly the same time, then the preterite ("began") would be appropriate.
Sorry if that all sounds a bit complicated; if there's anything that's not clear, pop another question in.
Many thanks for your help. Yes, I think I understand (just about haha).
I would like to thank you on our Facebook page (The Runcorn and District Historical Society) and in the local press when we hang the next plaque. I always like to thank everyone who gives their help in the process of making the heritage plaques.
Are you okay with this? What is the name of the forum?
I will word it something like this:
Thanks to Paul Matthews from the ????? Grammar Forum.
I don't know if the story of this plaque will make our local newspaper at the moment, but here is the link to the story on the Facebook page for the Runcorn & District Historical Society.