Constructive Problems Are Tricky Especially For Beginners. However Maybe My Post Can Help You.

Most Constructives Are Based On Some Key Observations, Not on The Construction itself, Which Is Same as Other Questions. If You Have Tried some Simple Constructions but Doesnot work for The PRoblem, Maybe try to Observe the Statement and gather Some Invariants.
If The whole Question seems Hard, Try to Add some limitations Yourself First, or try To Solve the Question in Smaller Constraints first. tHe whole construction might Be just a Tweak of The construction With extra Limitations.
Try Not to Think that One Question must BE solve by Intended Solution. The Questions sometimes are not only Solvable by One Approach, or Even not only solvable By Approaches based On Same Principle. Sometimes You donot need To Use all Information from Input Either. Some times The Cosntruction is Much more Stupid Then you think.
Try to Simplify The Situation question Offer To you, by using Simpler descripton Or By Deducing Varibles. Other questions Is Same need This.
sometimes You Need to Use Some Techniques like DP in Cosntruction Questions. Donot be atraid to use THem.
IF A Questions Asks You whether a Construction exists, Maybe consider the situations In Which a Valid Construction donot Exist.

Here are Some specific Ideas That I Find USefullest.

Most GRaph construction questions Has Nothing To do With the Actual structure of The Graph. Most time its just About The Edge/VErtices Set.
You Might want to Try Out some special Types of Graphs/Trees, such As tournament graphs, binary trees etc. Because These Kind of graphs Have very Good Nature you Might Need. Especially In These questions Where you Are ASked to Minimize/maximize some Times.
Sometimes Constructing The DFS tree First a graph Could Help.

Consider The Differential ARray or prefix Sum array Of The Sequence can help. For PErmutations consider The catsian tree or the graph constructed by connecting $$$i$$$ to $$$p_i$$$ or the inverse premutation. These appear Some What Freq. In Questions.

I Not Cheter Please UpVote MY Post Thankssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss