Japt -R, 20 18 14 bytes

21Æ`¬v`úX+7"ef

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21Æ`¬v`úX+7"ef
21Æ                :Map each X in the range [0,21)
   `¬v`            :  Compressed string "cov"
       ú           :  Pad left
        X+7        :    To length X+7
           "ef     :    With "ef"
                   :Implicit output joined with newlines

Perl 5, 39 bytes

$b=covfef;say$b.=substr$b,-2,1for 1..21

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Perl 5, 45 bytes

$_="covfef
";eval's/((.).)$/$1$2/;print;'x 20

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Python 3, 61 bytes

def f(n=20):
	print(' '*n+('covf'+'ef'*12)[:~n])
	if n:f(n-1)

not the shortest, but prints triangle without e/f vertical adjacency

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C (gcc), 77 75 74 bytes

f(i,j){for(i=4;i++<25;)for(printf("\ncov"),j=i;--j;)putchar("ef"[i-j&1]);}

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Japt, 17 16 bytes

4o25_îeif)i`¬v

Test it online! 14 bytes of code, +2 for -R flag. If we can output as an array, simply remove the -R and subtract 2 bytes.

Explanation

4o25              Generate the range [4, 5, ..., 24].
    _             Map each item in this range to
      eif           "fe" (really weird hack, but it works and saves a byte)
     î   )          repeated to length <item>
          i`¬v      with "cov" inserted at the beginning.
                  Implicit: output result of last expression, joined with newlines (-R)

Alternatively, you could do 7o28_îfie)h"cov, which does the same thing, except overwriting cov onto the beginning of the string instead of inserting it at the beginning.

Pyth, 29 bytes

Vr4 25+"cov"+*/N2"fe"?%N2"f"k

Explanation:

Vr 4 25      Loops variable N from 4 to 24 (excludes 25)
    */N2"fe"     Repeats string "fe" N / 2 number of times 
        ?%N2"f"k     Concatenates "f" if N % 2 is 1 (odd), or an empty 
                     string if it is even (N % 2 is 0) 
            +            Concatenates strings

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Retina, 27 bytes

covf12$*
1
ef
.
$`¶
G`.{7}

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covf12$*

Add covf plus 12 1s.

1
ef

Change each 1 to ef. This results in the last line of the desired output, but with an extra f.

.
$`¶

Replace the string with a list of all of its proper prefixes.

G`.{7}

Delete the prefixes that are too short.

Pyth, 33 bytes

V21+++"cov"*"fe"2*"fe"s/N2*\f%N2

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Husk, 15 bytes

↑21↓7ḣ+¨¢√¨¢"fe

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NB: ¨¢√¨ is the compressed strind "cov"

Jelly, 17 bytes

⁾feẋ12“cov”;;\ṫ7Y

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Python 2, 53 52 bytes

f=lambda n=21:n*' 'and'\ncovf'+('ef'*12)[:-n]+f(n-1)

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05AB1E, 17 16 bytes

„fe12×禦¦…covì»

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Explanation

„fe               # push the string "fe"
   12×            # repeat it 12 times
      η           # get the prefixes of this string
       ¦¦¦        # drop the first 3
          …covì   # prepend the string "cov" to each
               »  # join on newline

Charcoal, 13 bytes

Ｅ²¹⁺cov…fe⁺⁴ι

Try it online! Link is to verbose version of code. Explanation:

            Implicitly print joined with newlines
Ｅ²¹         the implicit range 0..21 mapped to
   ⁺cov     the concatenation of "cov" with
    …fe     the characters "fe" repeated until their length is exactly
     ⁺⁴ι    4 more than the value

Python 2, 37 bytes

s='covfef'
exec"s+=s[-2];print s;"*21

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Matches the text shown.

Python 2, 36 bytes

s='covfefe'
exec"print s;s+='?';"*21

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Prints:

covfefe
covfefe?
covfefe??
covfefe???
covfefe????
covfefe?????
covfefe??????
covfefe???????
covfefe????????
covfefe?????????
covfefe??????????
covfefe???????????
covfefe????????????
covfefe?????????????
covfefe??????????????
covfefe???????????????
covfefe????????????????
covfefe?????????????????
covfefe??????????????????
covfefe???????????????????
covfefe????????????????????

Totally valid answer which matches the specs/rules.

PowerShell, 25 Bytes. Try it online!

"covfefe";8..28|%{'?'*$_}

OR (same byte count)

0..21|%{'covfefe'+'?'*$_}

As intended answer which matches the example output.

PowerShell, 44 Bytes. Try it online!

$c="covfef";0..20|%{($c=$c+('e','f')[$_%2])} 

Ugly assignment at the beginning, i'm sure there's some that can be saved here.

C# (Mono), 132 bytes

using System.Linq;_=>{for(int i=0;i<21;)System.Console.WriteLine("cov"+string.Concat(new int[4+i++].Select((n,j)=>j%2<1?"f":"e")));}

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SOGL V0.12, 16 15 bytes

4'⁷Ν{ņcoļvƧfemo

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Explanation:

4'⁷Ν             push inclusive range from 4 to 24
    {            for each do
     ņcoļv         output in a new line "cov" (split into 2 commands of "output 2 chars in a new line" and "output next char" for a byte save)
          Ƨfemo    output "fe" molded into the length of the current item

Ruby - 53 chars

s='covfef';21.times{|i|puts s +=(i.odd? ? 'f' : 'e')}

Note: This is the OP's answer, there must be better one's out there.