There's nothing in the first paragraph that tells us whether it happened more than once, but in the second paragraph "always" tells us that it was something they did regularly.

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Certainly, I can help you identify the usage of "at" and "on" in the sentences, and clarify the "during" and "every year" aspect. In the provided text: "At new year," refers to a specific point in time when Lydia and her family hosted a small party. This is a particular moment or event. "On new year's day," signifies a specific day, in this case, New Year's Day. It's a day when they engage in a particular activity, such as going for a family walk. Regarding "during" and "every year": "During" is used when referring to a period of time in which an event happens. For example, "During the party, they ate homecooked food." "Every year" indicates that the action is a recurring event, happening annually. In this context, "On New Year's Day, they always went for a big family walk around a local country park" suggests that this is a tradition they follow each year. So, to summarize, "at" and "on" are used to pinpoint specific moments or days, and "during" and "every year" provide information about the timing and recurrence of events.