The phrase "slender thread" is a way of describing a very thin and delicate connection or link between things. It can suggest that something important or significant hinges on this fragile connection. Full definition
Jackson LJ went on to describe paragraphs 3.2.2.2 (ii) and 3b.5.1 of the Technical Requirements as «too slender a thread upon which to hang a finding that MTH gave a warranty of 20 years life for the foundations». (lawyer-monthly.com)
Elias LJ, with whom Lewison and Floyd LJJ agreed, said at [52] that s 23C: «[I] s an extremely slender thread on which to hang such extensive and burdensome duties... if Parliament had intended to confer a power of this scope, it would have done so expressly.» (newlawjournal.co.uk)
Let me suggest that the slender thread connecting Trump to the church is his occasional holiday appearances at Marble Collegiate Church, made famous by its pastor for 52 years, Norman Vincent Peale. (christianitytoday.com)