Gladys went to
Tonlé Sap, a lake in Cambodia, which also happens to be the biggest freshwater lake in Southeast Asia.

Marker placed by Wikipedia, relief map by User:NordNordWest
The crossword grid and cryptic clues:

Across
1. Pieces of trivia which may or may not be correct or false at disco production (8)
FACTOIDS (a factoid can mean either "a piece of trivia that is correct" or "a piece of trivia that is wrong") = F (false) ATDISCO*
6. People on the inside have transformed into statues, ultimately standing still (10)
MOTIONLESS = MOLES (people on the inside, spies) have INTO* + _S
7. Western county's ruthless ruler embraces end of democracy (4)
MAYO (western county in Ireland) = MAO (Mao Zedong, "ruthless ruler") embraces _Y
8. "Poor sap!" – psychiatrist's last words (4)
SPAT (words, as in a quarrel) = SAP* + _T
12. Small core of local team that fights the likes of Doctor Doom, Vulture and Wolverine? (10)
SCAVENGERS ("vulture and wolverine?") = S + (-lo)C(-al) + AVENGERS
13. Teacher from Ecuador unexpectedly has temperature (8)
EDUCATOR = ECUADOR* has T
Down
1. Building component for cell in a daft city construction (5,4)
FATTY ACID anagram
2. Cut length of synthetic rope (4)
CROP = hidden word ("length" as in "a length of rope")
3. Literally exhausted after working alone (4)
ONLY = L_Y after ON (working/activated, as in a device)
4. Leave one's application into German academy remotely, to start with (9)
DISAPPEAR = I'S APP into DE + A_ R_
5. "I am a Muslim leader!" (4)
IMAM &lit: I'M A M_
9. Cause a shock moving east (4)
TASE anagram
10. Submit to retreat? (4, abbr.)
EVAC (evacuation) &lit: CAVE reversed
11. Bond is moving west, intent on keeping prisoners (4)
KNOT = reversed hidden word
The Masyu:
We start with the circles near the edges, because it is clear how the loops pass through them.

It is not hard to see why the right side quickly becomes this (use the white circle and anti-loop rules):

And thus the lines through the black circles are forced. As one of them is in a blue square, the line passing through the circles must be blue.

Now, there is only one way that a line can pass through the circle in the red square in the bottom right:

Using the anti-loop rule and the rule that blue and red loops must be separate, we get:

The white circle immediately above the bottom end of the blue loop can only be passed through by the loop in one way. Also, the top end of the blue loop cannot pass through the white circle in the blue square because it breaks the white circle rule: the line will not have turned immediately on either side of the circle. So that end goes up, and it forces the line through the white circle in the blue square, and also the line through the two white circles to the left of that. That last line cannot go down to the black circle since again it breaks the white circle rule. So we get this:

And now the vertical part of the loop through the unpassed black circle is forced. Note that the end at the bottom has to turn according to the white circle rules.

Now that part of the loop cannot be blue, because this will trap one of the ends of the part of the red loop at the bottom; it cannot connect to another red part. So that is red. Hence the end of the loop that was “trapped” earlier now has one and only one way to connect to the black circle, according to black circle rules. And that also makes the two blue ends right above it connect:

There’s only one way for the red part of the loop at the bottom to proceed:

And the white circle beside the black circle in the red loop can only be passed by a line in one way. Because of this the line is forced to be red. It also forces the line through the two white circles beside the white circle in the blue square to be red. Then, the loop is forced to reach the top:

Let’s not forget the top left:

Which forces this:

Finally, we cannot reach the blue square and close the blue loop at the same time. But the rules don’t say that we have to pass through every colored cell, so we can close the loop like this and we are done.

After these,
We can overlay the Masyu loops on the crossword.

Note that there are eight letters not passed through by either loop, and these spell out the answer!