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I'm trying to avoid using an adverb in the following sentence:

When people asked him where the money was going, he [said frankly], "a good cause".

I'm hoping there's a synonym for "said" that elicits feelings of terseness and simplicity in the speaker's voice, but I can't think of one.

I want to know more about the character in your example to provide a good option because you could elicit a similar feeling using a different type of word altogether. Part of me would like to see:

When people asked him where the money was going, he quipped, "a good cause".

even though that is more witty. I also like the following, which, depending on the full context, could give off a similar "terse but honest" vibe:

...he shrugged, " a good cause".

...he leveled, "a good cause".

...he maintained, "a good cause".

I could go on, and, while I typed this out, some others have posted good answers. I'm a fan of context and character to choose the best "said" replacement.

Though none are quite synonymous with "saying frankly," a few words that carry a similar connotation are declared, avowed, professed, and affirmed.

I think the closest verb you will find is to avow.

He avowed the money was for a good cause.

I don't believe it is exactly synonymous with the phrasal "he stated frankly", as it seems also to suggest a sense of "promise" or "guarantee", rather than just straightness and sincerity. But the two are very close.

This particular sense of avow is number 5, of the OED listing of the verb.

5. transitive. To own, acknowledge, admit, or confess (facts, statements, or opinions, that one might himself conceal or deny).
   Const. as in 4. 1330 R. Mannyng Chron. 320 If he wille avowe alle
   his wikked sawe. c1386 Chaucer Canon's Yeoman's Prol. & Tale 89 I
   wol nought avowe what I say, And therfor kep it secré I yow pray.
   c1440 Promptorium Parvulorum 19 A-vowyn, or stonde by the forsayde
   worde or dede, Advoco. a1464 J. Capgrave Abbreuiacion of Cron.
   (Cambr. Gg.4.12) (1983) 141 Sche accused a certeyn knyte, whech cam
   to þe kyng and avowid euery word. 1660 R. Allestree Gentlemans
   Calling ix. 159 This little Tract..must avow to come upon that
   uncivil, yet friendly errand. 1667 Dryden Annus Mirabilis 1666
   cxxxvii. 35 As when Fiends did Miracles avow, He stands confess'd
   ev'n by the boastful Dutch. 1778 Johnson in Boswell Life Johnson
   (1816) III. 294 Many a man thinks, what he is ashamed to avow. 1855
   W. H. Prescott Hist. Reign Philip II of Spain I. ii. viii. 235 The
   boldness with which he avowed his opinions.

According to Merriam-Webster, one sense of level
is “to deal frankly and openly.”

I found a use that seems to illustrate that:

He listened to our story with every sign of sympathy, and a touch of
sorrow. Then he leveled with us: "I think you're absolutely in the
right," he said, "but it's a political matter. I wouldn't touch it
with a ten-foot pole."

Odd Man Out: A Memoir of the Hollywood Ten, by Edward Dmytryk, Page 50.

And (again according Merriam-Webster), senses of confide
include “to show confidence by imparting secrets” and “to tell confidentially.”

And a use that seems illustrative to me:

... [He] was very well approved of by his master, who
found him daily more and more intelligent in all things relating to
traffic, and so confided in him that he sent him with some ships...

Rascals in Paradise, by James A. Michener, ‎Page 76