The OP asks two questions wrapped into one phrase. "Is there an objective *thing*?", and "Is there an objective *truth*?" 

Question 1 is about the existence of a mind-independent reality made of (mind-independent) "objects". IMO it's quite straightforward to answer by the affirmative. Most of us conclude very early on in our development, around the age of 2 or so, that objects tend to continue to exist even when you don't see them and don't mind their existence. It's called "object permanence".

There's a bit of a complication when one realizes that objects are always distinguished from other objects, and defined or conceived as different from other objects, by a subject. Supposedly, without subjects trying to analyse the world and categorize the stuff around them, there would be still an objective reality "out there", but not necessarily segmented, or contrasted the way we see it. For all we know, the universe may be just one big whole, every thing connected to everything else. So there's a valid argument that we (subjects) do the segmenting of the universe into distinct objects by looking at it analytically. In this sense, objects are not totally mind-independent: their are always *objects of one's attention* in a way. But they are mind-independent enough, in the sense that once you delineated them as distinct from the rest, objects behave mind-independently. "The mountain" will be here tomorrow and the day after, reliably so, even though an objective limit is hard to define between "the mountain" and "the valley.

Some objects -- usually smaller than mountains, say one's keys -- can get lost. They can "go missing", that is. When that happens, most people do *not* assume that their keys have just vanished into thin air while they were not being looked at. Even philosophers, even *idealist philosophers* look for their keys when their keys "go missing". They look in their pockets, in yesterday's trousers, in the washing machine, etc. and often enough, ultimately they do find them.

Even when they don't find them, most folks, even some solipsists I know, tend to think: "I must have misplaced them" or "something happened to them", something consistent with the principle of object permanence, that is. Rare are those who would conclude that their keys have just stopped existing. 

Even things that we would *wish* stopped existing tend to keep on existing, against our wish. Object permanence (and hence some degree of mind-independence) does seem to hold up, as a general primciple. 

Question 2 is a bit thornier. In summary, "truth" is a property of statements, specifically of those statements that pretend to describe the world or part of it, like "the cat is on the mat", aka so-called synthetic statements. If the statement accurately describes or corresponds to a given state of affairs, then it's deemed true. 

(At least that's in broad brush one of the meanings of the word "truth")

The problem with speaking of "objective truth" in that sense of the word "truth" is that the word "objective" then must mean something else than "mind independent", because a statement by definition needs to be stated, otherwise it's not a statement. A statement is always stated *by someone*, some locutor, and usually for the benefit of someone else, for some audience. So a statement cannot be seen as mind-independent: by necessity, it comes from a mind and goes to another. 

Fortunately, that's not really what people mean when they say things like "it's an objective truth that the cat is on the mat." They don't mean that "the cat is on the mat" is a statement stated by no one and for the benefit of no one. Instead they mean that *anyone* and *everyone* put in the right context (in that room with the cat) would agree that there appears to be some cat on some mat. 

Objectivity, when applied to statements, generally implies near-universal *intersubjective agreement* about some statement describing some (mind-independent enough) state of affairs.

And in that sense of the word, there *are* objective truths. Perhaps surprisingly, people do tend to agree on factual statements about cats being located on mats. It's a pitty they disagree about pretty much everything else.