I didn't suggest this earlier, because you wanted to keep the constant term while dropping non-constant terms below a threshold. Your last comment suggests that you do want to zero-out the linear term. So, if you wanted to drop all terms below a threshold, including the constant term, then you could use Series as follows:

Normal @ Series[1/x^3 + 1/x^2 + 1/x + 1 + x + x^2 + x^3 + x^4 + x^5, {x, Infinity, -3}]]
(* x^3 + x^4 + x^5 *)

If you wanted to drop terms below 1/x, then:

Normal @ Series[1/x^3 + 1/x^2 + 1/x + 1 + x + x^2 + x^3 + x^4 + x^5, {x, Infinity, 1}]]
(* 1 + 1/x + x + x^2 + x^3 + x^4 + x^5 *)

I would recommend using CoefficientRules and FromCoefficientRules:

CoefficientRules[2 + 3 x - x^2 + 5 x^3 + 9 x^8]
DeleteCases[%, ({n_} -> _) /; 1 <= n < 4]
FromCoefficientRules[%, x]

Or if you prefer function based, rather than rule-based:

CoefficientRules[2 + 3 x - x^2 + 5 x^3 + 9 x^8]
Select[%, Not[1 <= #[[1, 1]] < 4] &]
FromCoefficientRules[%, x]

Thanks for the reply and its good sense approach.

Since my earlier method to strike that subpolynomial took too long, Tech Support used zero replacement to swiftly strike it. But then I noticed an occasional wrong result caused by a linear term c x that replacement didn't touch. Unable to zero out the subpolynomial-with-linear-term, I turned to Wolfram Community to see if Replacement could be made to do so.

In the actual application, the full polynomial (in x) is the sum of many bilinear forms a.B.c where B is the power of another matrix whose entries are powers of x -- B also depends upon the summation index (whose values are tuples). And so computing that subpolynomial isn't so easy. Although it seems that Replacement can't be made to behave as I had hoped, your idea to use Coefficient is a good one that I'll remember.