You have publishing pages that exist in a Pages library within a sub-site. You have set up site permissions so that (for example) User1 has Read permission at the site collection level, has NO permission at the sub-site or Pages Library level, but has Read permission to individual pages in that Pages library. That user tries to browse to one of the individual pages that contains a Summary Links control and receives an "Access Denied" message.
This is expected behavior at the time of this article's creation. When loading a Publishing page with a Summary Links control on it, SharePoint tries to load Event Receivers which results in "Access Denied" as Event Receivers are present at the Pages Library level where this user doesn’t have access.
In other words: despite of having Read access at the individual page level, the user still gets "Access Denied" because they have no permission to run the Event Receivers that are loaded when the page is rendered.
Grant the affected User 'Read' permissions at Pages library level
Create a new Permission Level (which is a subset of Read Permissions) with below Permissions. Make the affected user a part of that new Permission Level and add the new Permission Level to the 'Pages' Library.